---
abstract: 'New graded modules for the current algebra of ${\mathfrak{sl}}_n$ are introduced. Relating these modules to the fusion product of simple ${\mathfrak{sl}}_n$-modules and local Weyl modules of truncated current algebras shows their expected impact on several outstanding conjectures. We further generalize results on PBW filtrations of simple ${\mathfrak{sl}}_n$-modules and use them to provide decomposition formulas for these new modules in important cases.'
address:
- 'Mathematisches Institut, Universität zu Köln, Germany'
- 'School of Mathematics and Statistics, University of Glasgow, UK'
author:
- Ghislain Fourier
bibliography:
- 'biblist-fusion.bib'
title: 'New homogeneous ideals for current algebras: filtrations, fusion products and Pieri rules'
---

[^1]

Introduction
============

We consider the simple complex Lie algebra ${\mathfrak{sl}}_n = {\mathfrak{b}} \oplus {\mathfrak{n}}^-$ and its current algebra ${\mathfrak{sl}}_n \otimes \bc[t]$. We fix a pair $(\lambda_1, \lambda_2)$ of dominant integral ${\mathfrak{sl}}_n$-weights. $F_{\lambda_1, \lambda_2}$ will be introduced as the cyclic ${\mathfrak{sl}}_n \otimes \bc[t]$-module defined by the homogeneous ideal generated by the kernel of an evaluation map of ${\mathfrak{b}} \otimes \bc[t]$ and certain monomials in $U({\mathfrak{n}}^- \otimes \bc[t])$. $F_{\lambda_1, \lambda_2}$ decomposes into simple, finite-dimensional ${\mathfrak{sl}}_n$-modules: $$F_{\lambda_1, \lambda_2} =_{{\mathfrak{sl}}_n} \bigoplus_{\tau \in P^+} V(\tau)^{\oplus a_{\lambda_1, \lambda_2}^{\tau}}.$$ As $F_{\lambda_1, \lambda_2}$ is a highest weight module, we have $a_{\lambda_1, \lambda_2}^{\lambda_1 + \lambda_2} = 1 \text{ and } a_{\lambda_1, \lambda_2}^{\tau} = 0 \text{ if } \tau \nleq \lambda_1 + \lambda_2.$ Moreover, ${\mathfrak{sl}}_n \otimes t^2 \bc[t].F_{\lambda_1, \lambda_2} = 0$ and hence $$F_{\lambda_1, \lambda_2} =  U({\mathfrak{n}}^- \otimes \bc[t]/(t^2)).\mathbbm{1} \cong U({\mathfrak{n}}^-) S({\mathfrak{n}}^-).\mathbbm{1}.$$ Due to this observation, the ${\mathfrak{sl}}_n$-highest weight vectors and therefore the multiplicities $ a_{\lambda_1, \lambda_2}^{\tau}$ should be “controlled“ by $S({\mathfrak{n}}^-).\mathbbm{1}$. This provides a close relation to the framework of PBW filtrations ([@FFoL11a; @FFoL13a]). By construction, $F_{\lambda_1, \lambda_2}$ is a quotient of $S({\mathfrak{n}}^-)/\mathcal{I}(\lambda_1, \lambda_2)$ with an induced ${\mathfrak{n}}^+$-action $\circ$, where the ideal is generated by $$U({\mathfrak{n}}^+) \circ \langle f_\alpha^{a_\alpha + 1}\, | \, \text{ for all positive roots } \alpha \rangle \subset S({\mathfrak{n}}^-)$$ for some $a_\alpha$ depending on $\lambda_1, \lambda_2$. Generalizing the results from [@FFoL11a Theorem and Theorem B] we see that a spanning set of $S({\mathfrak{n}}^-).\mathbbm{1}$ can be parameterized by integer points in a polytope defined through Dyck paths conditions (Corollary \[span-m\]). This leads to the question wether one can give a polytope parametrizing the highest weight vectors. We give a positive answer in certain important cases:

\[intro-thm\] Suppose $\lambda_1, \lambda_2 $ satisfy one of the following:

1.  $\lambda_1, \lambda_2$ are both rectangular weights, e.g. multiples of some fundamental weights $\omega_i, \omega_j$,

2.  $\lambda_1$ is arbitrary and $\lambda_2$ is either $\omega_j$ or $k \omega_1$,

3.  $\lambda_1 + w(\lambda_2)$ is dominant for all Weyl group elements,

then for all dominant weights $\tau$: $$a_{\lambda_1, \lambda_2}^{\tau} =  c_{\lambda_1, \lambda_2}^{\tau}, \text{  the Littlewood-Richardson coefficients.}$$

Part (2) might be seen as a Pieri rule while part (3) covers $\lambda_1 \gg \lambda_2$. So for fixed $\lambda_2$ we cover the minimal case, e.g. $\lambda_1$ being a fundamental weight, and the large case, e.g. $\lambda_1 \gg \lambda_2$. Note that the results from [@CV13], [@V13] imply $a_{\lambda_1, \lambda_2}^{\tau} =  c_{\lambda_1, \lambda_2}^{\tau}$ for all $\tau \in P^+$ if $\lambda_1 = m \omega_i$ and the height of $\lambda_2$ is less than $m+1$. This covers of course part (1) of the theorem but we provide here a different proof using the relation to the PBW filtration.\
The paper is motivated by the search for homogeneous ideals in $U({\mathfrak{sl}}_n \otimes \bc[t])$ defining the fusion product $V(\lambda_1)_{c_1} \ast V(\lambda_2)_{c_2}$ of two simple ${\mathfrak{sl}}_n$-modules. This is the associated graded module of the tensor product of corresponding evaluation modules ([@FL99] and also Section \[fusion\]). These ideals can be deduced straightforward for ${\mathfrak{sl}}_2$ (Lemma \[fusion-sl2\]) and we generalize this to obtain generators for every ${\mathfrak{sl}}_2$-triple. The theorem implies that in the considered cases (Lemma \[whatever\]) $$F_{\lambda_1, \lambda_2} \cong V(\lambda_1)_{c_1} \ast V(\lambda_2)_{c_2}$$ and we conjecture that this is true for all pairs of dominant integral weights.\
Let us briefly explain why these modules $F_{\lambda_1, \lambda_2}$ and especially the conjectured isomorphism to the fusion product is of special interest. In fact, this is closely related to several important conjectures:\
The first one is the conjecture that the fusion product of finitely many tensor factors is independent of the evaluation parameter ([@FL99]). This independence has been proved for some classes of modules but so far not for arbitrary tuples of dominant integral weights. Note that two-factor case can be deduced from straightforward calculations.\
The second conjecture is on Schur positivity of certain symmetric functions. In [@CFS14] (see also [@DP2007]) a partial order on pairs of dominant weights has been introduced. It is conjectured that along with the partial order, the difference of the products of the corresponding Schur functions is a non-negative linear combination of Schur functions (this has been conjectured also independently by Lam, Postnikov and Pylyavskyy), hence *Schur positive*. Note here, that this generalizes a conjecture on Schur positivity along row shuffles ([@Oko97; @FFLP05]), proved in [@LPP07].\
The third related conjecture is on local Weyl modules for truncated current algebras. Local Weyl modules for generalized current algebras, ${\mathfrak{sl}}_n \otimes A$, where $A$ is a commutative, associative, unital $\bc$-algebra, have gained much attention in the last two decades. Due to their homological properties they play an important role in the category of finite-dimensional ${\mathfrak{sl}}_n \otimes A$-modules, which is not semi-simple in general (for more see [@CFK10]).\
Although quite a lot of research has been done on local Weyl modules, their explicit character is known for a few algebras only. For $A = \bc[t^{\pm 1}], \bc[t]$ their character is given by the tensor product of fundamental modules for ${\mathfrak{sl}}_n$ ([@CL06; @FoL07]), for semi-simple, finite-dimensional $A$, the character is given by $\dim A$ copies of a simple ${\mathfrak{sl}}_n$-module. Besides these cases the character is not known for general local Weyl modules, not even for the “smallest“ non-semi-simple algebra $A = \bc[t]/(t^2)$.\
It is conjectured that for $A = \bc[t]/(t^K)$ this character is also, similar to $\bc[t]$, given by the tensor product of simple ${\mathfrak{sl}}_n$-modules. We investigate here on the $K=2$ case and prove that in this case the local Weyl modules are isomorphic to certain $F_{\lambda_1, \lambda_2}$, more detailed: $(\lambda_1, \lambda_2)$ is the unique maximal element in the aforementioned poset of pairs of dominant weights (adding up to a fixed $\lambda$).\
A proof, that the $a_{\lambda_1, \lambda_2}^{\tau}$ are in fact the Littlewood-Richardson coefficients, would imply the conjectures on Schur positivity and on local Weyl modules immediately (Lemma \[fusion-sp\], Lemma \[weyl-f\]) and gives another proof for the two-factor of the independence conjecture (Lemma \[whatever\]).

The paper is organized as follows: In Section \[one\] we refer to the basic definitions and in Section \[graded\] we introduce the modules $F_{\lambda_1, \lambda_2}$, proving first properties. In Section \[PBW\] we recall the PBW filtration and work out the relation to our new modules, while in Section \[fusion\] we recall the fusion products and work out their relation to our modules. In Section \[firstproof\] we give the proof for the ${\mathfrak{sl}}_2$ -case and part(3) of Theorem \[intro-thm\]. Section \[KR\] contains the proofs of part (1) of Theorem \[intro-thm\] and Section \[pieri\] the proof of part (2). Section \[poset\] recalls the partial order on pairs of dominant weights and also local Weyl modules, and relates these constructions to the new modules.

**Acknowledgement:** The author would like to thank Evgeny Feigin for various discussions on these modules and explaining the calculations for the independence conjecture in the two-factor case, and further Christian Korff for asking about Pieri rules.

Preliminaries {#one}
=============

Let ${\mathfrak{g}} = {\mathfrak{sl}}_{n}(\bc)$, the special linear Lie algebra. We fix a triangular decomposition ${\mathfrak{sl}}_n = {\mathfrak{n}}^+ \oplus {\mathfrak{h}} \oplus {\mathfrak{n}}^-$ and denote a fixed set of simple roots $\Pi = \{ \alpha_1, \ldots, \alpha_{n-1}\}$, here we use the numbering from [@Bou02]. Further, we set $I =\{ i, \ldots, n-1 \}$. The sets of roots is denoted R, the set of positive roots $R^+$. Every root $\beta \in R^+$ can be expressed uniquely as $\alpha_i + \alpha_{i+1} + \ldots + \alpha_j$ for some $i \leq j$, we denote this root $\alpha_{i,j}$. For $\alpha \in R$, we denote the root space $${\mathfrak{g}}_{\alpha} = \{ x \in {\mathfrak{sl}}_n \, | \, [h,x] = \alpha(h) x \; \forall \; h \, \in {\mathfrak{h}} \} = \langle x_\alpha^+ \, | \, \text{ if } \alpha \in R^+ \rangle.$$ Further, for $\alpha \in R^+$, we fix a ${\mathfrak{sl}}_2$-triple $\{ x_\alpha^+, x_{\alpha}^-, h_{\alpha} \}$. Denote $P \subset {\mathfrak{h}}^*$, respective $P^+$ the integral weights, respective dominant integral weights, and $\{ \omega_1, \ldots, \omega_{n-1}\}$ the set of fundamental weights.\

We recall some notations and facts from representation theory. Let $V$ be a finite-dimensional ${\mathfrak{sl}}_{n}$-module, then $V$ decomposes into its weight spaces with respect to the ${\mathfrak{h}}$-action $$V = \bigoplus_{ \tau \in P } V_\tau =  \bigoplus_{ \tau \in P }  \{ v \in V \; | \; h.v = \tau(h).v \text{ for all } h \in {\mathfrak{h}}\}$$ $P^+$ parameterizes the simple finite-dimensional modules. For $\lambda \in P^+$ we denote the simple, finite-dimensional ${\mathfrak{sl}}_n$-module of highest weight $\lambda$ by $V(\lambda)$. Further we denote by $v_{\lambda}$ a highest weight vector of $V(\lambda)$

The category of finite-dimensional ${\mathfrak{sl}}_n$-modules is semi-simple, hence the tensor product of two simple modules decomposes into the direct sum of simple modules, so for $\lambda_1, \lambda_2 \in P^+$ $$V(\lambda_1) \otimes V(\lambda_2) \cong_{{\mathfrak{sl}}_n} \bigoplus_{\tau \in P^+} V(\tau)^{c_{\lambda_1, \lambda_2}^\tau}.$$ Here, $c_{\lambda_1, \lambda_2}^\tau$ denotes the multiplicity of the simple module $V(\tau)$ in a decomposition of the tensor product. These numbers are known as Littlewood-Richardson coefficients and there are several known formulas to compute them ([@Kli68; @N2002; @Lit94] to name but a few).

The vector space ${\mathfrak{sl}}_n \otimes \bc[t]$ equipped with the bracket $$[x \otimes p(t) , y \otimes q(t)] = [x,y] \otimes p(t)q(t) \; \; \forall \, x,y \in {\mathfrak{sl}}_n, p(t), q(t) \in \bc[t]$$ is a Lie algebra and called the *current algebra* of ${\mathfrak{sl}}_n$. One may also view this as the Lie algebra of regular functions on $\bc$ with values in ${\mathfrak{sl}}_n$ (see [@NSS12]). The natural grading on $\bc[t]$ induces a grading on $U({\mathfrak{sl}}_n \otimes \bc[t])$, where the component of degree $0$ is $U({\mathfrak{sl}}_n \otimes 1)$.

For a fixed $k \geq 1$, the *truncated current algebra* is the graded quotient of the current algebra $${\mathfrak{sl}}_n \otimes \bc[t] / ({\mathfrak{sl}}_n \otimes t^K \bc[t]) \cong {\mathfrak{sl}}_n \otimes \bc[t]/(t^K).$$ In this paper we will be dealing mainly with the $K=2$ case. Then $U({\mathfrak{sl}}_n \otimes \bc[t]/(t^2))$ can be seen as the smash product of $U({\mathfrak{sl}}_n)$ and the polynomial ring $S({\mathfrak{n}}^-)$ ([@Hag13]).

The representation theory of ${\mathfrak{sl}}_n \otimes \bc[t]$ has been subject to a lot of research during the last 25 years ([@CP01; @FF02; @CM04; @CL06; @FoL06; @FoL07; @Nao12] to name but a few). The most important property we should mention is, that the category of finite-dimensional ${\mathfrak{sl}}_n \otimes \bc[t]$-modules is not semi-simple.\
Every simple, finite-dimensional module is the tensor product of evaluation modules ([@Rao93]). This is still true if we replace $\bc[t]$ by a commutative, finitely generated algebra $A$ and instead of complex numbers, evaluations in pairwise distinct maximal ideals [@CFK10].\
Although the simple, finite-dimensional modules are therefore easily described and quite well understood, the task of understanding the indecomposable modules is still unsolved (besides the case $A = \bc[t], \bc[t^{\pm}]$ and the cases where $A$ is finite-dimensional and semi-simple).\
Even in the case where $A$ is the two-dimensional truncated polynomial ring, $A = \bc[t]/(t^2)$, the category of finite-dimensional modules is far from being well understood. While the simple modules are in one-to-one correspondence to simple modules of ${\mathfrak{sl}}_{n}$ (by using the evaluation at the unique maximal ideal of $ \bc[t]/(t^2)$, [@CFK10]), there is not much known about indecomposables, projectives etc. We will return to this point in Section \[poset\].

Some new graded module {#graded}
======================

We introduce new graded modules for ${\mathfrak{sl}}_{n}  \otimes \bc[t]$ as follows. For fixed $\lambda_1, \lambda_2 \in P^+$, let $$(\lambda_1 + \lambda_2) : {\mathfrak{h}} \longrightarrow \bc_{\lambda_1 + \lambda_2}$$ be the one-dimensional ${\mathfrak{h}}$-module. We extend this trivially to an action of ${\mathfrak{b}} = {\mathfrak{n}}^+ \oplus {\mathfrak{h}}$ on $\bc_{\lambda_1 + \lambda_2}$. And further, by evaluation at $t = 0$, we obtain a one-dimensional module $${\mathfrak{b}} \otimes \bc[t] \longrightarrow {\mathfrak{b}} \longrightarrow {\mathfrak{h}} \longrightarrow \bc_{\lambda_1 + \lambda_2}.$$ We consider the induced module for the subalgebra $({\mathfrak{b}} \otimes 1) \oplus ({\mathfrak{sl}}_n \otimes t\bc[t]) \subset {\mathfrak{sl}}_n \otimes \bc[t]$ $$\Ind_{{\mathfrak{b}} \otimes \bc[t]}^{{\mathfrak{b}} \otimes 1 \oplus {\mathfrak{sl}}_n \otimes t\bc[t]} \bc_{\lambda_1 + \lambda_2}$$ and denote by $M_{\lambda_1, \lambda_2}$ the quotient by the left ideal generated by $${\mathfrak{n}}^- \otimes t^2\bc[t] \text{ and }  (f_{\alpha} \otimes t)^{\min \{\lambda_1(h_\alpha), \lambda_2(h_\alpha) \} + 1 }, \, \forall \; \alpha \in R^+.$$ We introduce the ${\mathfrak{sl}}_n \otimes \bc[t]$-module $F_{\lambda_1, \lambda_2}$ as the maximal integrable (as a ${\mathfrak{sl}}_n$-module) quotient $$F_{\lambda_1, \lambda_2} = \overline{\Ind_{{\mathfrak{b}} \otimes 1 \oplus {\mathfrak{sl}}_n \otimes t\bc[t]}^{{\mathfrak{sl}}_n \otimes \bc[t]} M_{\lambda_1, \lambda_2}}.$$

Due to the construction, we can give defining relations on a generator of $F_{\lambda_1, \lambda_2}$.

\[relation\] Let $\lambda_1, \lambda_2 \in P^+$ and $\lambda = \lambda_1 +  \lambda_2$. Then $F_{\lambda_1, \lambda_2}$ is the ${\mathfrak{sl}}_n \otimes \bc[t]$-module generated through $w$ with relations $${\mathfrak{n}}^+ \otimes \bc[t].w = 0 , \ {\mathfrak{h}} \otimes t \bc[t].w = 0 , \ {\mathfrak{n}}^- \otimes t^2\bc[t] = 0$$ and for all $\alpha \in R^+$ and $h \in {\mathfrak{h}}$: $$0 = (f_{\alpha} \otimes 1)^{\lambda(h_\alpha) + 1}.w = (f_\alpha \otimes t)^{\min\{\lambda_1(h_\alpha), \lambda_2(h_\alpha) \}+1} = (h\otimes 1 - \lambda(h)).w.$$

We have to deal with the ${\mathfrak{sl}}_n$-relation only. But since $F_{\lambda_1, \lambda_2}$ is integrable we have immediately $(f_\alpha \otimes 1)^{\lambda(h_\alpha) + 1}.\mathbbm{1} = 0$. Therefore $F_{\lambda_1, \lambda_2}$ is a quotient of the module given by the relations in the proposition. On the other hand, every module satisfying the relations is an integrable quotient of $\Ind_{{\mathfrak{b}} \otimes 1 \oplus {\mathfrak{sl}}_n \otimes t\bc[t]}^{{\mathfrak{sl}}_n \otimes \bc[t]} M_{\lambda_1, \lambda_2}$.

\[firstprop\] Let $\lambda_1, \lambda_2 \in P^+$. Then

1.  $F_{\lambda_1, \lambda_2}$ is a non-negatively graded ${\mathfrak{sl}}_{n} \otimes \bc[t]$-module.

2.  $F_{\lambda_1, \lambda_2}$ is finite-dimensional.

3.  $F_{\lambda_1, \lambda_2}= \bigoplus_{s \geq 0} F_{\lambda_1, \lambda_2}^s$, and $ F_{\lambda_1, \lambda_2}^s$ is a ${\mathfrak{sl}}_n$-module.

4.  $F_{\lambda_1, \lambda_2}$ has a unique simple quotient isomorphic to $V(\lambda_1 + \lambda_2)_0$.

5.  ${\mathfrak{sl}}_{n} \otimes t^2 \bc[t]. F_{\lambda_1, \lambda_2} = 0$, and hence $F_{\lambda_1, \lambda_2}$ is a ${\mathfrak{sl}}_n \otimes \bc[t]/(t^2)$-module.

Part $(1)$ is clear, since the defining relations of $F_{\lambda_1, \lambda_2}$ are homogeneous and $U({\mathfrak{sl}}_n \otimes \bc[t])$ is non-negatively graded. Due to the defining relations, $F_{\lambda_1, \lambda_2}$ is a quotient of the local graded Weyl module for $U({\mathfrak{sl}}_n \otimes \bc[t])$ of highest weight $\lambda_1 + \lambda_2$, $W_{\bc[t]}(0, \lambda_1 + \lambda_2)$ (see Proposition \[f-weyl\] or [@CP01] for details, they are not relevant here). In [@CP01] it is shown that this local graded Weyl module is finite-dimensional, which implies $(2)$.\
Now, as ${\mathfrak{sl}}_n \cong {\mathfrak{sl}}_n \otimes 1 \hookrightarrow {\mathfrak{sl}}_n \otimes \bc[t]$, $F_{\lambda_1, \lambda_2}$ is also a finite-dimensional ${\mathfrak{sl}}_n$-module, hence decomposes into a direct sum of simple finite-dimensional ${\mathfrak{sl}}_n$-modules. Moreover, as $U({\mathfrak{sl}}_n)$ is the degree $0$ part of $U({\mathfrak{sl}}_n \otimes \bc[t])$, we see that each graded component $F^s_{\lambda_1, \lambda_2}$ is a ${\mathfrak{sl}}_n$-module and each simple ${\mathfrak{sl}}_n$-module is contained in a unique $F_{\lambda_1, \lambda_2}^s$. This implies $(3)$.\
The degree $0$ component of $F_{\lambda_1, \lambda_2}$ is obviously isomorphic to $V(\lambda_1 + \lambda_2)$ as a ${\mathfrak{sl}}_n$-module. A standard argument shows that $$U({\mathfrak{sl}}_n \otimes \bc[t]){\mathfrak{sl}}_n \otimes t\bc[t].\mathbbm{1}$$ is the maximal proper submodule not containing $\mathbbm{1}$. The quotient by this submodule is isomorphic to the graded evaluation module $V(\lambda_1 + \lambda_2)_0$, this gives $(4)$. Part $(5)$ follows again immediately from the defining relations.

Since $F_{\lambda_1, \lambda_2}^s$ is a ${\mathfrak{sl}}_{n}$-module, it has a decomposition into a direct sum of simple ${\mathfrak{sl}}_{n}$-modules $$F_{\lambda_1, \lambda_2}^s \cong_{{\mathfrak{sl}}_{n}} \bigoplus_{\tau \in P^+} V(\tau)^{\oplus a_{\lambda_1, \lambda_2}^\tau(s)}, \text{ for some } a_{\lambda_1, \lambda_2}^{\tau}(s) \geq 0.$$ We set $$a_{\lambda_1, \lambda_2}^{\tau} :=  \sum\limits_{s \geq 0} a_{\lambda_1, \lambda_2}^{\tau}(s),$$ so we have $$\dim \Hom_{{\mathfrak{sl}}_{n}}( F_{\lambda_1, \lambda_2}, V(\tau)) =  a_{\lambda_1, \lambda_2}^{\tau} .$$

We see immediately from Proposition \[relation\]:

\[cor-hw\] Let $\lambda_1, \lambda_2 \in P^+$, then $$a_{\lambda_1, \lambda_2}^{\lambda_1 + \lambda_2} = 1 \, ; \, a_{\lambda_1, \lambda_2}^{\tau} = 0 \text { for } \tau \nleq \lambda_1 + \lambda_2.$$

The main theorem of the paper is the following:

\[main-thm\] Let $\lambda_1, \lambda_2\in P^+$, then we have $ a_{\lambda_1, \lambda_2}^{\tau} \geq c_{\lambda_1, \lambda_2}^{\tau}$, $\forall \; \tau \in P^+$.\
Moreover:

1.  (Pieri rules) Let $\lambda_1 \in P^+$, $\lambda_2 \in \{ \omega_j, k \omega_1\}$ for some $j \in I$ or $k  \geq 1$, then: $a_{\lambda_1, \lambda_2}^{\tau} = c_{\lambda_1, \lambda_2}^{\tau}$, $\, \forall \, \tau \in P^+$.

2.  Let $\lambda_1 = m_i \omega_i, \lambda = m_j \omega_j$ for some $i, j \in I, m_i, m_j \geq 0$, then: $a_{\lambda_1, \lambda_2}^{\tau} = c_{\lambda_1, \lambda_2}^{\tau}$, $\, \forall \, \tau \in P^+$.

3.  If $\lambda_1 \gg \lambda_2$, then: $a_{\lambda_1, \lambda_2}^{\tau} = c_{\lambda_1, \lambda_2}^{\tau}$, $\, \forall \, \tau \in P^+$.

The proofs will be given in the following sections, but we should note the following here:

\[first-rem\] In the proof we will see that $\lambda_1 \gg \lambda_2$ can be made precise, by requesting $$c_{\lambda_1, \lambda_2}^{\tau} = \dim V(\lambda_2)_{\tau - \lambda_1}$$ for all $\tau \in P^+$. Note that this is equivalent to $\lambda_1 + w(\lambda_2) \in P^+$ for all $w \in W$, the Weyl group of ${\mathfrak{sl}}_n$.

From the work [@CV13; @V13] one can deduce further that $a_{\lambda_1, \lambda_2}^{\tau} = c_{\lambda_1, \lambda_2}^{\tau}$ if $\lambda_1 = m \omega_i$ and $\lambda_2(h_\theta) \leq m$ (where $\theta$ is the highest root of ${\mathfrak{sl}}_n$). The authors were using relations on Demazure modules and their fusion products, generalizing an approach presented in [@FoL06]. This of course includes $(2)$ of the theorem but we give a new proof here that might be generalized to other but rectangular weights.

PBW filtration and polytopes {#PBW}
============================

In this section we recall the PBW filtration and we will see how the results from [@FFoL11a] can be adapted here in order to understand the ${\mathfrak{sl}}_{n}$-structure on $F_{\lambda_1, \lambda_2}$.\
By the PBW theorem and the construction of $F_{\lambda_1, \lambda_2}$ as an induced module we know that $$F_{\lambda_1, \lambda_2} = U({\mathfrak{n^-}})U({\mathfrak{n^-}} \otimes t). \mathbbm{1}.$$ In order to understand the ${\mathfrak{sl}}_{n}$-decomposition of $ F_{\lambda_1, \lambda_2}$ it would be sufficient to parametrize all ${\mathfrak{sl}}_{n}$-highest weight vectors. The equation above suggests, that this set of highest weight vectors should be controlled by $U({\mathfrak{n^-}} \otimes t). \mathbbm{1}$. We start with analyzing this.

We recall the notion of Dyck path from [@FFoL11a]:\
A Dyck path of length $s$ is a sequence of positive roots $$\mathbf{p} = (\beta_1, \ldots, \beta_s)$$ with $s\geq 1$ and such that if $\beta_i = \alpha_{k,\ell}$ then $\beta_{i+1} \in \{ \alpha_{k+1, \ell}, \alpha_{k, \ell +1} \}$. If $\beta_1 = \alpha_{k_1, \ell_1}$ and $\beta_s = \alpha_{k_s, \ell_s}$, then we call $$\alpha_{k_1, \ell_s} \text{ the base root of the path } \mathbf{p}, \text{ denoted by } \beta(\mathbf{p})$$ Denote the set of all Dyck paths by $\mathbb{D}$.

The PBW filtration on $U({\mathfrak{n}}^-)$ is given as follows: $$U({\mathfrak{n}}^-)^{\leq s} = \left\langle x_1 \cdots x_r \, | \, 0 \leq r\leq s \text{ and } x_i \in {\mathfrak{n}}^-  \right\rangle_{\bc}$$ The associated graded algebra is a commutative algebra isomorphic to $S({\mathfrak{n}}^-)$, the polynomial ring in ${\mathfrak{n}}^-$. The adjoint action of ${\mathfrak{n}}^+$ on ${\mathfrak{sl}}_n$ induces an action $\circ$ on $S({\mathfrak{n}}^-)$.\
We fix a tuple of non-negative integers $$\mathbf{a} := (a_\alpha) \in \bz_{\geq 0}^{n(n-1)/2}$$ and consider the ideal $\mathcal{I}(\mathbf{a}) \subset S({\mathfrak{n}}^-)$ given by $$\mathcal{I}(\mathbf{a}) = S({\mathfrak{n}}^-)\left\langle \sum_{\alpha \in R^+}  U({\mathfrak{n}}^+) \circ f_\alpha^{a_{\alpha}+1} \right\rangle.$$

We fix $\mathbf{a} = (a_\alpha)$ and define a polytop in $ \mathbb{R}^{n(n-1)/2}$: $$\mathcal{P}(\mathbf{a}) = \left\{ (x_\alpha) \in \mathbb{R}^{n(n-1)/2} \, | \, \forall \; \mathbf{p} \in \mathbb{D}: \sum_{ \alpha \in \mathbf{p}} x_\alpha \leq a_{\beta(\mathbf{p})}\right\} .$$ We denote $$S(\mathbf{a}) = \mathcal{P}(\mathbf{a}) \cap  \mathbb{Z}_{\geq 0}^{n(n-1)/2}$$ the set of integer points in $ \mathcal{P}(\mathbf{a})$.\
This construction of the polytope covers the cases considered in [@FFoL11a; @FFoL11b; @FFoL13; @FFoL13a; @G11; @BD14], where $a_\alpha := \lambda(h_\alpha)$ for some fixed $\lambda \in P^+$.\
We define further the degree and the weight of an integer point: Let $\mathbf{s} = (s_\alpha) \in  \mathbb{Z}_{\geq 0}^{n(n-1)/2}$, then $$\deg (\mathbf{s}) = \sum_{\alpha \in R^+} s_\alpha \text{ and }  {\operatorname{wt}}(\mathbf{s}) = \sum_{\alpha \in R^+} s_\alpha \alpha \in P.$$

Although, our approach generalizes the construction provided in [@FFoL11a], we obtain a similar result on a spanning set of $S({\mathfrak{n}}^-)/ \mathcal{I}(\mathbf{a})$ (see [@FFoL11a Theorem 2]). For this denote $$\mathbf{f}^{\mathbf{t}} = \prod_{\alpha \in R^+} f_\alpha^{t_\alpha} \in S({\mathfrak{n}}^-) \text{ where } \mathbf{t} = (t_\alpha) \in  \mathbb{Z}_{\geq 0}^{n(n-1)/2}.$$

\[span-set\] We fix $\mathbf{a} = (a_\alpha) \in  \mathbb{Z}_{\geq 0}^{n(n-1)/2}$, then $$\{ \overline{\mathbf{f}^{\mathbf{s}}} \; | \;\mathbf{s} \in S(\mathbf{a}) \}$$ is a spanning set of $S({\mathfrak{n}}^-)/ \mathcal{I}(\mathbf{a})$.

Here we follow the idea in [@FFoL11a]. ${\mathfrak{n}}^+$ acts by differential operators on $S({\mathfrak{n}}^-)$, namely $e_\alpha \circ f_\beta = f_{\beta - \alpha}$ or $0$ if $\beta - \alpha$ is a positive root. Using these differential operators and an appropriate total order $\prec$ on the monomials in $S({\mathfrak{n}}^-)$, we can prove in exactly the same way as [@FFoL11a Proposition 1] a straightening law. Namely if $\mathbf{s} \notin S(\mathbf{a})$, then $$\overline{\mathbf{f}^{\mathbf{s}}} = \overline{\sum_ { \mathbf{t} \prec \mathbf{s} } c_{\mathbf{t}} \mathbf{f}^{\mathbf{t}}}.$$ This implies now the lemma. For more details we refer to [@FFoL11a].

In [@FFoL11a], and the case $a_\alpha := \lambda(h_\alpha)$, for some fixed $\lambda \in P^+$, it was further proved that this set is in fact a basis. We can not prove this here and although we conjecture that this is also true in our generality.

By construction $M_{\lambda_1, \lambda_2}$ is a cyclic $U({\mathfrak{n^-}} \otimes t)$-module. So there exists an ideal $\mathcal{I}_{\lambda_1, \lambda_2}$ such that $$M_{\lambda_1, \lambda_2} \cong U({\mathfrak{n^-}} \otimes t)/\mathcal{I}_{\lambda_1, \lambda_2}.$$ Since $M_{\lambda_1, \lambda_2}$ is a ${\mathfrak{n}}^+$-module, the ideal $\mathcal{I}_{\lambda_1, \lambda_2}$ is stable under the adjoint action of ${\mathfrak{n}}^+$ (on $U({\mathfrak{n}}^- \otimes t)$). Moreover the action is a graded action (where ${\mathfrak{n}}^+$ has degree $0$). Note that we have the identification $$S({\mathfrak{n}}^-) \cong U({\mathfrak{n}}^-  \otimes t) \subset U({\mathfrak{sl}}_n \otimes \bc[t]/(t^2))  \text{ via } \mathbf{f}^{\mathbf{t}} \mapsto \prod_{\alpha} (f_\alpha \otimes t)^{t_\alpha}.$$ Then we have the obvious proposition:

\[prop-maps\] For $\lambda_1, \lambda_2 \in P^+$ we set $$\mathbf{a} := (a_\alpha) \text{ where } a_\alpha = \min\{\lambda_1(h_\alpha), \lambda_2(h_\alpha)\}.$$ Then we have maps of $S({\mathfrak{n}}^-)$-modules $$S({\mathfrak{n}}^-) / \mathcal{I}(\mathbf{a}) \twoheadrightarrow M_{\lambda_1, \lambda_2} \twoheadrightarrow U({\mathfrak{n}}^- \otimes t).\mathbbm{1} \subset F_{\lambda_1, \lambda_2}.$$

To emphasize the dependence on $\lambda_1, \lambda_2$, we denote the set of integer points $S(\mathbf{a})$ in this case by $S(\lambda_1, \lambda_2)$. Then, combining Proposition \[prop-maps\] and Lemma \[span-set\], we have:

\[span-m\] Fix $\lambda_1, \lambda_2 \in P^+$, then $$\{ \mathbf{f}^{\mathbf{s}}.\mathbbm{1} \; | \; \mathbf{s} \in S(\lambda_1, \lambda_2) \}$$ is, via the identification, a spanning set for $M_{\lambda_1, \lambda_2}$ and hence for $U({\mathfrak{n}}^- \otimes t).\mathbbm{1} \subset F_{\lambda_1, \lambda_2}$.

In order to identify the ${\mathfrak{sl}}_n$-highest weight vectors in $F_{\lambda_1, \lambda_2}$ with images of $\prod_{\alpha} (f \otimes t)^{s_\alpha} .\mathbbm{1}$ for some $\mathbf{s} \in S(\lambda_1, \lambda_2)$, we introduce an appropriate filtration of $U({\mathfrak{n}}^- \otimes t)$. First we filter by the degree of $t$ and the further by the height of the weights. Finally, we filter further by a total order on the monomials.\
Recall that $U({\mathfrak{n}}^- \otimes t) \cong S({\mathfrak{n}}^-)$ if considered as the subalgebra in $U({\mathfrak{sl}}_n \otimes \bc[t]/t^2)$ as we continue to do. Therefore $U({\mathfrak{n}}^- \otimes t)$ is naturally graded by $t$ and we keep denoting the graded components $U({\mathfrak{n}}^- \otimes t)^s$. For $\tau \in P$, we denote $$U({\mathfrak{n}}^- \otimes t)_\tau = \{ v  \in  U({\mathfrak{n}}^- \otimes t) \ | \ {\operatorname{wt}}(v) = \tau \}.$$ All weights of $U({\mathfrak{n^-}} \otimes t)$ are in $\bigoplus_{i  \in I} \bz_{\leq 0} \alpha_i$. Let $\tau = \sum_{i \in I} a_ i \alpha_I  \in \bigoplus_{i  \in I} \bz_{\leq 0} \alpha_i$. Then we denote the height of $\tau$ $$\hei(\tau) := \sum_{i \in I} - a_i.$$ So we have a filtration of the graded components $$U({\mathfrak{n}}^- \otimes t)^{s, \leq \ell} = \langle  u \in  U({\mathfrak{n}}^- \otimes t)^s_\tau \ | \ \hei(\tau) \leq \ell \}.$$ This is spanned by monomials of total degree $s$ and whose weights have height less or equals to $\ell$.\
On the other hand, $U({\mathfrak{n}}^- \otimes t)$ is $\bz_{\geq 0}^{n(n-1)/2}$ graded. Each graded component is one-dimensional, spanned by $\prod_{\alpha \in R^+} (f_\alpha \otimes t)^{s_\alpha}$ for some $\mathbf{s} \in \bz_{\geq 0}^{n(n-1)/2}$. We order the $n(n-1)/2$-tuples by first ordering the positive roots $$\alpha_{i,j} \leq \alpha_{k,\ell} :\Leftrightarrow i <  k \text{ or }  i = k \text{ and } j \leq \ell .$$ Using the lexicographic order $\leq$ we obtain an order on the monomials spanning $U({\mathfrak{n}}^- \otimes t)$.\
Combining this we introduce a finer filtration on $U({\mathfrak{n}}^- \otimes t)^s$. So given $s, \ell \geq 0$ and $\mathbf{n}\in \bz_{\geq 0}^{n(n-1)/2}$ with $\deg (\mathbf{n}) = s$, $\hei(-{\operatorname{wt}}(\mathbf{n})) = \ell$, we have $$U({\mathfrak{n}}^- \otimes t)^{s, \leq \ell}_{\leq  \mathbf{n}} =  U({\mathfrak{n}}^- \otimes t)^{s, < \ell} + \left\langle \prod_{\alpha \in R^+} (f_\alpha \otimes t)^{m_\alpha} \ | \ \deg(\mathbf{m}) = s \, , \, \hei( -{\operatorname{wt}}\mathbf{m}) = \ell \, , \, \mathbf{m} < \mathbf{n} \right\rangle_{\bc}.$$

We turn back to the module $F_{\lambda_1, \lambda_2}$ and recall its graded components $F^s_{\lambda_1, \lambda_2}$. We define $$\mathcal{F}^{\leq \ell}(F_{\lambda_1, \lambda_2}^s) := U({\mathfrak{sl}}_n) U({\mathfrak{n}}^- \otimes t)^{s, \leq \ell} .\mathbbm{1} \subseteq F_{\lambda_1, \lambda_2}^{s}.$$ By construction $$\mathcal{F}^{\leq \ell}(F_{\lambda_1, \lambda_2}^s) /\mathcal{F}^{ < \ell}(F_{\lambda_1, \lambda_2}^s)$$ is a ${\mathfrak{sl}}_n$-module and we have the following

Let $\mathbf{s} \in S(\lambda_1, \lambda_2)$, then the image of $$\mathbf{f}^{\mathbf{s}}.\mathbbm{1} \in \mathcal{F}^{\leq \ell}(F_{\lambda_1, \lambda_2}^s) /\mathcal{F}^{< \ell}(F_{\lambda_1, \lambda_2}^s)$$ is either $0$ or a ${\mathfrak{sl}}_n$-highest weight vector of weight $\lambda_1 + \lambda_2 - {\operatorname{wt}}(\mathbf{s} )$.

Since $\hei(e_\beta) > 0 $, we see using the commutator relations that $$e_{\beta} \prod_{\alpha}(f_\alpha \otimes t)^{s_{\alpha}} \in U({\mathfrak{n}}^- \otimes t)^{s, \leq \ell} U({\mathfrak{n}}^+)_+ + U({\mathfrak{n}}^- \otimes t)^{s, < \ell} U({\mathfrak{n}}^+).$$ This implies that $$e_{\alpha} \prod_{\alpha}(f_\alpha \otimes t)^{s_{\alpha}} = 0 \in \mathcal{F}^{\leq \ell}(F_{\lambda_1, \lambda_2}^s) /\mathcal{F}^{< \ell}(F_{\lambda_1, \lambda_2}^s).$$

We see that, by choosing this appropriate filtration, the highest weight vectors (for the ${\mathfrak{sl}}_n$-action) of the associated graded module $F_{\lambda_1, \lambda_2}$, are of the form $\mathbf{f}^{\mathbf{s}}.\mathbbm{1}$ for some $\mathbf{s}$.\
By using the refinement of the filtration we can say even more. So given $s, \ell \geq 0$ and $\mathbf{n}\in \bz_{\geq 0}^{n(n-1)/2}$ with $\deg (\mathbf{n}) = s$, $\hei(-{\operatorname{wt}}(\mathbf{n})) = \ell$, we have $$\mathcal{F}_{\leq \mathbf{n}}^{\leq \ell}(F_{\lambda_1, \lambda_2}^s) := U({\mathfrak{sl}}_n) U({\mathfrak{n}}^- \otimes t)_{\leq \mathbf{n}}^{s, \leq \ell} .\mathbbm{1} \subset F_{\lambda_1, \lambda_2}^{s}.$$ Then the graded components $$\mathfrak{G}_{\mathbf{n}}^{s, \ell}(F_{\lambda_1, \lambda_2})  := \mathcal{F}_{\leq \mathbf{n}}^{\leq \ell}(F_{\lambda_1, \lambda_2}^s) / \left( \mathcal{F}^{<  \ell}(F_{\lambda_1, \lambda_2}^s) + \sum_{ \mathbf{m} < \mathbf{n} } \mathcal{F}_{\leq \mathbf{m}}^{\leq  \ell}(F_{\lambda_1, \lambda_2}^s) \right)$$ are simple ${\mathfrak{sl}}_n$-modules.

We have seen in Corollary \[span-m\] that the monomials corresponding to points in $S(\lambda_1, \lambda_2)$ are a spanning set of $U({\mathfrak{n}}^{-} \otimes t). \mathbbm{1}$.

We say $\mathbf{n} \in S(\lambda_1, \lambda_2)$ is a *highest weight point* if $\mathfrak{G}_{ \mathbf{n}}^{s, \ell}(F_{\lambda_1, \lambda_2}) $ is non-zero for $s = \deg(\mathbf{n})$ and $\ell = \hei(-{\operatorname{wt}}(\mathbf{n}))$. The set of highest weight points is denoted $S_{hw}(\lambda_1, \lambda_2)$.

Note that, since $F_{\lambda_1, \lambda_2}$ is an integrable ${\mathfrak{sl}}_n$-module, we have for all $\mathbf{s} \in S_{hw}(\lambda_1, \lambda_2)$ $$\lambda_1  + \lambda _ 2 - {\operatorname{wt}}(\mathbf{s}) \in P^+.$$

\[cor-upper\] For $\lambda_1, \lambda_2, \tau \in P^+$ we have $$\dim \Hom_{{\mathfrak{sl}}_n} (F_{\lambda_1, \lambda_2}, V(\tau)) = \sharp \{  \mathbf{s} \in S_{hw}(\lambda_1, \lambda_2) \, | \, {\operatorname{wt}}(\mathbf{s}) = \lambda_1 + \lambda_2 - \tau \}$$ Moreover $$\dim \Hom_{{\mathfrak{sl}}_n} (F^s_{\lambda_1, \lambda_2}, V(\tau)) = \sharp \{  \mathbf{s} \in S_{hw}(\lambda_1, \lambda_2) \, | \, {\operatorname{wt}}(\mathbf{s}) = \lambda_1 + \lambda_2 - \tau \text{ and } \deg( \mathbf{s}) = s.\}$$

Fusion products {#fusion}
===============

In this section we recall the fusion product of two simple ${\mathfrak{sl}}_{n}$-modules and work out the relation to the modules $F_{\lambda_1, \lambda_2}$.

The following construction is due to [@FL99]. Recall the grading on $U({\mathfrak{sl}}_{n} \otimes \bc[t])$ given by the degree function on $\bc[t]$ $$U({\mathfrak{sl}}_n \otimes \bc[t]) ^r = \{ u \in U({\mathfrak{sl}}_n \otimes \bc[t]) \; | \; \deg(u) \leq r \}.$$ Then $\mathcal{F}^{0} = U({\mathfrak{sl}}_n)$ and we set $\mathcal{F}^{-1} = 0$.\
Let $V(\lambda_1), \ldots, V(\lambda_k)$ be simple ${\mathfrak{sl}}_n$-modules of highest weights $\lambda_1, \ldots, \lambda_k$. Further let $c_1, \ldots, c_k$ be pairwise distinct complex numbers. Then $V(\lambda_i)$ can be endowed with the structure of a ${\mathfrak{sl}}_n \otimes \bc[t]$-module via $$x \otimes p(t).v = p(c_i)x.v \; \; \text{ for all } x \in {\mathfrak{sl}}_n, p(t) \in \bc[t], v \in V(\lambda_i),$$ we denote this module $V(\lambda_i)_{c_i}$. Then $$V(\lambda_1)_{c_1} \otimes \cdots \otimes V(\lambda_k)_{c_k}$$ is cyclic generated by the tensor product of highest weight vectors $v_{\lambda_1} \otimes \cdots \otimes v_{\lambda_k}$ (even more it is simple [@Rao93; @CFK10]). The grading on $U({\mathfrak{sl}}_n \otimes \bc[t])$ induces a filtration on $V(\lambda_1)_{c_1} \otimes \cdots \otimes V(\lambda_k)_{c_k}$ $$U({\mathfrak{sl}}_n \otimes \bc[t])^{\leq r}. v_{\lambda_1} \otimes \cdots \otimes v_{\lambda_k}.$$ Since $U({\mathfrak{sl}}_n \otimes \bc[t])$ is graded, the associated graded is again a module for $U({\mathfrak{sl}}_n \otimes \bc[t])$, denoted usually by $$V(\lambda_1)_{c_1} \ast \cdots \ast V(\lambda_k)_{c_k},$$ and is called the *fusion product*. Recall that the graded components are $U({\mathfrak{sl}}_n)$-modules, since $U({\mathfrak{sl}} \otimes 1)$ is the degree $0$ component of $U({\mathfrak{sl}}_n \otimes \bc[t])$. Further, since we have not changed the ${\mathfrak{sl}}_n$-structure in this construction:

\[fusion-decom\] Let $\lambda_1, \lambda_2 \in P^+$, $c_1 \neq c_2 \in \bc$, then for all $\tau \in P^+$ $$\dim \Hom_{{\mathfrak{sl}}_n} (V(\lambda_1)_{c_1} \ast V(\lambda_2)_{c_2}, V(\tau)) = c_{\lambda_1, \lambda_2}^{\tau}.$$

\[whatever\] For $\lambda_1, \lambda_2 \in P^+$, $c_1 \neq c_2 \in \bc$ we have a surjective map of ${\mathfrak{sl}}_{n} \otimes \bc[t]$-modules: $$F_{\lambda_1, \lambda_2} \twoheadrightarrow V(\lambda_1)_{c_1} \ast V(\lambda_2)_{c_2},$$ moreover $a_{\lambda_1, \lambda_2}^{\tau} \geq c_{\lambda_1, \lambda_2}^{\tau}$, $ \forall \, \tau \in P^+$.

We prove the ${\mathfrak{sl}}_2$-case first. Here dominant integral weights are parameterized by $\bz_{\geq 0}$, and for $k \geq 0$ let $V(k) =Sym^k \bc^2$. Fix $k \geq m \geq 0$. Then $$\dim (V(k)_{c_1} \otimes V(m)_{c_2})_{k + m - 2 \ell} = \begin{cases} \ell + 1 \text{ for } 0 \leq \ell \leq m \\ m + 1 \text{ for } m \leq \ell \leq  k \\  k + m + 1- \ell \text{ for } k \leq \ell \leq k + m \end{cases}$$ Since $c_1 \neq c_2$, we se, using the Vandermonde determinant, that $$(f_\alpha \otimes t)^{m+1} v_k \otimes v_m \in \langle (f_\alpha \otimes 1)^{m+1} v_k \otimes v_m, \ldots ,(f_\alpha \otimes 1) (f_\alpha \otimes t)^m v_k \otimes v_m \rangle_{\bc}$$ since the $k-2$-weight space is at most $m+1$-dimensional. This implies that $(f_\alpha \otimes t)^{m+1} v_k \otimes v_m$ is $0$ in the associated graded module.\
We see further, that the weight space of weight $k+m -2$ is two dimensional and spanned by the vectors $ (f_\alpha \otimes 1) v_k \otimes v_m, (f_\alpha \otimes t) v_k \otimes v_m$. This implies that for $\ell \geq 2$, $(f_\alpha \otimes t^\ell) v_k \otimes v_m = 0$ in the fusion product, similarly we see that for all $\ell \geq 1$, $h \otimes t^\ell v_k \otimes v_m = 0$ in the fusion product.\
This implies that there is a surjective map of ${\mathfrak{sl}}_2$-modules $$F_{k\omega,m\omega} \twoheadrightarrow V(k)_{c_1} \ast V(m)_{c_2}.$$ Let us turn to the general case. Let $\lambda_1, \lambda_2 \in P^+$, $c_1 \neq c_2 \in \bc$, $\alpha \in R^+$, and let $m = \min \{ \lambda_1(h_\alpha), \lambda_2(h_\alpha) \}$. By considering the ${\mathfrak{sl}}_2$-triple $\{ e_\alpha, h_\alpha, f_\alpha\}$ we see with the same argument as above that $$(f_\alpha \otimes t)^{m+1} v_{\lambda_1} \otimes v_{\lambda_2} \in \operatorname{ span } \{  (f_\alpha \otimes 1)^{m+1} v_{\lambda_1} \otimes v_{\lambda_2}, \ldots , (f_\alpha \otimes 1)(f_\alpha \otimes t)^m v_{\lambda_1} \otimes v_{\lambda_2} \}$$ This implies that $ (f_\alpha \otimes t)^{m+1} v_{\lambda_1} \otimes v_{\lambda_2} = 0$ in the associated graded. The remaining defining relations for $F_{\lambda_1, \lambda_2}$ are easily verified.

Using this lemma we have the following very interesting consequence:

If $\forall \, \tau \in P^+$: $a^{\tau}_{\lambda_1, \lambda_2} = c^{\tau}_{\lambda_1, \lambda_2}$, then for all $c_1 \neq c_2 \in \bc:$ $$V(\lambda_1)_{c_1} \ast V(\lambda_2)_{c_2} \cong_{{\mathfrak{sl}}_{n} \otimes \bc[t] } F_{\lambda_1, \lambda_2}.$$ Moreover, the fusion product in this case is independent of the parameter $c_1, c_2$, providing another proof of a conjecture by B.Feigin and S.Loktev ([@FL99]).

By Lemma \[whatever\] we have for all $\lambda_1, \lambda_2 \in P^+$ and $c_1 \neq c_2 \in \bc$ a surjective map of ${\mathfrak{sl}}_n \otimes \bc[t]$-modules $$F_{\lambda_1, \lambda_2} \twoheadrightarrow V(\lambda_1)_{c_1} \ast V(\lambda_2)_{c_2}.$$ With Corollary \[fusion-decom\] we know that the multiplicity of $V(\tau)$ in the fusion product is $c_{\lambda_1, \lambda_2}^\tau$. By assumption, this is equal to $a_{\lambda_1, \lambda_2}^\tau$, which is the multiplicity of $V(\tau)$ in $F_{\lambda_1, \lambda_2}$. So the modules are isomorphic as ${\mathfrak{sl}}_n$-modules and hence by a dimension argument also as ${\mathfrak{sl}}_n \otimes \bc[t]$-modules.\
Since $F_{\lambda_1, \lambda_2}$ is a graded module and independent of any evaluation parameter, the same is true for the fusion product $V(\lambda_1)_{c_1} \ast V(\lambda_2)_{c_2}$.

First proofs for parts of the main theorem {#firstproof}
==========================================

We prove here the ${\mathfrak{sl}}_2$-case, namely $a_{m \omega_1, k \omega_1}^{\tau} = c_{m \omega_1, k \omega_1}^{\tau}$ for all $m, k \geq 0$ and $\tau \in P^+$. In the following section we prove the $\lambda_1 \gg \lambda_2$-case.

In this section we consider the ${\mathfrak{sl}}_2$-case. In this case, dominant integral weights are parametrized by non-negative integers.

\[fusion-sl2\] Let $m_1, m_2 \geq 0$, then for all $c_1 \neq c_2 \in \bc$ $$F_{m_1 \omega_1, m_2 \omega_1} \cong_{{\mathfrak{sl}}_2 \otimes \bc[t]} V(m_1 \omega_1)_{c_1} \ast V(m_2 \omega_1)_{c_2}.$$ Moreover, $a_{m_1 \omega_1, m_2 \omega_1}^{k \omega_1} = c_{m_1 \omega_1, m_2 \omega_1}^{k \omega_1}$.

This proves Theorem \[main-thm\](1) for $A_1$.

Let $m_1, m_2 \in \mathbb{Z}_{\geq 0}$, then by Lemma \[whatever\] it suffices to prove that $ \forall \; k \geq 0$ $$a_{m_1 \omega_1, m_2 \omega_1}^{k \omega_1} \leq  c_{m_1 \omega_1, m_2 \omega_1}^{k \omega_1} \;$$ Suppose $m_1 \geq m_2$. Then the relations of $F_{m_1 \omega_1, m_2 \omega_2}$ can be rewritten as $$(h \otimes 1).\mathbbm{1} = (m_1 + m_2 + 1). \mathbbm{1}; \; ; (f \otimes 1)^{m_1 + m_2 + 1}. \mathbbm{1} = 0 \; ; \; (f \otimes t)^{m_2 + 1}. \mathbbm{1}= 0,$$ while $({\mathfrak{n}}^- \otimes t^2 \bc[t] \oplus {\mathfrak{b}} \otimes t\bc[t] \oplus {\mathfrak{n}}^+ \otimes 1).  \mathbbm{1} = 0$. By considering $F_{m_1 \omega_1, m_2 \omega_1}$ as an ${\mathfrak{sl}}_2$-module we see from the relations, that it is generated by $$\{ \mathbbm{1}, (f \otimes t).\mathbbm{1}, (f \otimes t)^{2}.\mathbbm{1}, \ldots, , (f \otimes t)^{m_2}.\mathbbm{1} \} .$$ This implies that $F_{m_1 \omega_1, m_2 \omega_1}$ is multiplicity free and moreover we see that $$a_{m_1 \omega_1, m_2 \omega_1}^{k \omega_1} = 1 \Rightarrow k = m_1 + m_2 - 2 \ell \text{ for some } \ell \in \{0, \ldots, m_2 \}.$$ The famous Clebsch-Gordan formula gives for $k = m_1 + m_2 - 2 \ell \text{ for some } \ell \in \{0, \ldots, m_2 \}$ $$c_{m_1 \omega_1, m_2 \omega_1}^{k \omega_1} =1  \text{ and } c_{m_1 \omega_1, m_2 \omega_1}^{k \omega_1} = 0 \text{ else  }.$$ This implies (with Lemma \[whatever\]) $$a_{m_1 \omega_1, m_2 \omega_1}^{k \omega_1} \leq  c_{m_1 \omega_1, m_2 \omega_1}^{k \omega_1} \leq a_{m_1 \omega_1, m_2 \omega_1}^{k \omega_1}.$$

Note here, that this elementary result follows also from [@FF02] and [@CV13].

Let $\lambda_1, \lambda_2 \in P^+$. We say $$\lambda_1 \gg \lambda_2 \Leftrightarrow \lambda_1 + w(\lambda_2) \in P^+ \Leftrightarrow c_{\lambda_1, \lambda_2}^{\tau} = \dim V(\lambda_2)_{ \tau - \lambda_1},
\, \forall \tau \in P^+.$$ This is certainly satisfied if $\lambda_1(h_\alpha) \gg \lambda(h_\alpha)$ for all $\alpha \in R^+$.\
Suppose now $\lambda_1 \gg \lambda_2$, then $\min\{ \lambda_1(h_\alpha), \lambda_2(h_\alpha) \} = \lambda_2(h_\alpha)$. Which implies that if we define $$\mathbf{a} \in \bz^{n(n-1)/2} \text{ via } a_\alpha = \min \{ \lambda_1(h_\alpha), \lambda_2(h_\alpha) \}$$ then $a_\alpha = \lambda_2(h_\alpha)$. Let us denote $V(\lambda_2)^{a}$ the associated graded module obtained through the PBW filtration $U({\mathfrak{n}}^-)$ on the highest weight vector $v_{\lambda_2} \in V(\lambda_2)$ (see [@FFoL11a] for more details). This is a module for $S({\mathfrak{n}}^-)$, the associated graded algebra of $U({\mathfrak{n}}^-)$.

\[classicalpbw\] If $\lambda \gg \lambda_2$, then $$S({\mathfrak{n}}^-)/\mathcal{I}(\mathbf{a}) \cong V(\lambda_2)^{a}.$$

This is nothing but [@FFoL11a Theorem A].

We are ready to prove:

If $\lambda_1 \gg \lambda_2$, then $$a_{\lambda_1, \lambda_2}^{\tau} = c_{\lambda_1, \lambda_2}^{\tau}$$ and $$F_{\lambda_1, \lambda_2} \cong_{{\mathfrak{sl}}_n \otimes \bc[t]} V(\lambda_1)_{c_1} \ast V(\lambda_2)_{c_2}$$ for all $c_1 \neq c_2 \in \bc$.

With Corollary \[cor-upper\] we see that $$a_{\lambda_1, \lambda_2}^{\tau} \leq  \sharp \{  \mathbf{s} \in S(\lambda_1, \lambda_2) \, | \, {\operatorname{wt}}(\mathbf{s}) = \lambda_1 + \lambda_2 - \tau \}.$$ On the other hand, by Lemma \[whatever\], we have $$a_{\lambda_1, \lambda_2}^{\tau} \geq c_{\lambda_1, \lambda_2}^{\tau}.$$ By assumption $\lambda_1 \gg \lambda_2$, which implies (Remark \[first-rem\]) $$c_{\lambda_1, \lambda_2}^{\tau} = \dim V(\lambda_2)_{\tau - \lambda_1}.$$ Now [@FFoL11a Theorem B] gives in this case a parametrization of a basis of $V(\lambda_2)$ in terms of (in our notation) $S(\lambda_1, \lambda_2)$, namely $$\dim V(\lambda_2)_{\tau - \lambda_1} =  \sharp \{  \mathbf{s} \in S(\lambda_1, \lambda_2) \, | \, {\operatorname{wt}}(\mathbf{s}) = \lambda_2 - (\tau -\lambda_1)\}.$$ Which implies also $$a_{\lambda_1, \lambda_2}^{\tau} \leq c_{\lambda_1, \lambda_2}^{\tau},$$ hence the equality follows.

Rectangular weights {#KR}
===================

In this section we prove generators and relations for the fusion product of two arbitrary Kirillov-Reshetikhin modules. These modules are defined in the context of simple, finite-dimensional modules for the quantum affine algebra. They are indexed by a node $i \in I$, a level $m$ and an evaluation parameter $a \in \bc(q)^*$ and denoted $KR( m \omega_i, a)$. For more on their importance we refer here to the survey [@CH10].\
In this paper we consider the non-quantum analog (obtained through the $q\mapsto 1$ limit). In the ${\mathfrak{sl}}_n$-case, they are isomorphic to evaluation modules $V(m \omega_i)_{c}$ for some $c \in \bc$.\
We have seen in Lemma \[whatever\] that $$F_{m_i \omega_i, m_j \omega_j} \twoheadrightarrow V(m_i \omega_i)_{c_1} \ast V(m_j \omega_j)_{c_2}$$ for all $c_1 \neq  c_2$. We want to prove that this map is in fact an isomorphism, so we have to show that for all $\tau \in P^+$ $$a_{m_i \omega_i, m_j \omega_j}^{\tau} = c_{m_i \omega_i, m_j \omega_j}^{\tau} .$$

First, we will give formulas for the right hand side. We refer here to [@N2002] where the decomposition of a tensor product was computed by using combinatorics of Young tableaux. A formula for the tensor product of $V(\lambda_1)$ with $V(\omega_1)$ is given explicitly and as well as the induction procedure for $V(\lambda_2)$. In the special case of $\lambda_1 = m_i \omega_i$ and $\lambda_2 = m_j \omega_j$ one can deduce straightforward that for all $\tau \in P^+$: $$c_{m_i \omega_i, m_j \omega_j}^{\tau} \in \{ 0,1\}.$$ Moreover

For $i \leq j$, $c_{m_i \omega_i, m_j \omega_j}^{\tau}=1$ if and only if (setting $\omega_n = \omega_0 = 0$.) $$\tau = m_i \omega_i + m_j \omega_j + \sum_{q \geq 0}^{ \min\{ i, j+i, n-j\}} b_q(\omega_{i -q} + \omega_{j + q} -  \omega_i - \omega_j)$$ with $$\sum_{q \geq 0}^{ \min\{ i, j+i, n-j\}} b_q \leq \min\{ m_i, m_j\} \; \; , \; b_q \geq 0$$

Second, we will compute $a_{m_i \omega_i, m_j \omega_j}^{\tau}$. For this we identify again $$\mathbf{f}^{\mathbf{s}} \leftrightarrow \prod_{\alpha } (f_{\alpha} \otimes t)^{s_\alpha}.$$ Recall, from Section \[PBW\] (and [@FFoL11a]) that ${\mathfrak{n}}^+$ acts by differential operators on $S({\mathfrak{n}}^-)$. Here, we introduce a new class of operators as follows. Let $R^+_{\lambda_1, \lambda_2} = \{ \alpha \in R^+ \, | \, \lambda_1(h_\alpha) = \lambda_2(h_\alpha) = 0$}. Then ${\mathfrak{n}}^-_{\lambda_1, \lambda_2} = \langle f_\alpha \, | \, \alpha \in R^+_{\lambda_1, \lambda_2}\rangle $ is a subalgebra. We define for $\alpha \in R^+_{\lambda_1, \lambda_2}, \beta \in R^+$: $$f_\alpha \circ f_\beta \otimes t = \begin{cases} f_{\alpha +  \beta} \otimes t  \text{ if } \alpha + \beta \in R^+ \\ 0  \; \; \; \text{ else }  \end{cases}$$ This is induced by the adjoint action of ${\mathfrak{n}}^-$ on ${\mathfrak{n}}^- \otimes t$ (we normalize if necessary here). Moreover

This action induces an action of differential operators on $U({\mathfrak{n}}^- \otimes t).\mathbbm{1} \subset F_{\lambda_1, \lambda_2}$.

This follows easily from the fact that ${\mathfrak{n}}^-_{\lambda_1, \lambda_2}.\mathbbm{1} = 0 \in  F_{\lambda_1, \lambda_2}$.

In the following we will abbreviate $f_{\alpha_{k, \ell}}$ with $f_{k, \ell}$, $s_{\alpha_{k, \ell}}$ with $s_{k, \ell}$. Denote further $\mathbf{e}_{k,l}$, the basis vector of $\mathbb{R}^{n(n-1)/2}$ having $1$ for $e_{\alpha_{k, \ell}}$ and $0$ elsewhere. So let $\alpha \in R^+_{\lambda_1, \lambda_2}$ and $\gamma = \alpha + \beta \in R^+$, then $$f_\alpha \circ \mathbf{f}^{\mathbf{e}_\beta} = \mathbf{f}^{\mathbf{e}_\gamma}.$$

We turn to the case $\lambda_1 = m_i \omega_i, \lambda_2 = m_j \omega_j$. Let $\mathbf{s} \in S(\lambda_1, \lambda_2)$, then $s_{k, \ell} = 0$ for $\ell < j$ or $k > i$. The following is the crucial lemma, which gives an upper bound for the set of highest weight points.

\[diagonal\] Let $i\leq j \in I, m_i, m_j \geq 0$, and $p := \min \{i-1, n-1-j \}$, then $$U({\mathfrak{n}}^- \otimes t).\mathbbm{1} \subset U({\mathfrak{n}}^-) \langle (f_{i,j} \otimes t)^{a_0} (f_{i-1, j+1} \otimes t)^{a_1} \cdots (f_{i- p, j+p})^{a_k}. \mathbbm{1} \, | \,  a_q \geq 0, \forall  \,q \rangle.$$ Moreover we have $$S_{hw}(\lambda_1, \lambda_2) \subseteq \{ \mathbf{s} \in S(m_i \omega_i, m_j \omega_j) \, | \, s_{k, \ell} = 0 \text{ if } (k, \ell) \neq (i-q, j+q) \text{ for some } q \}.$$

We have seen in Corollary \[span-m\], that $$\{ \mathbf{f}^{\mathbf{s}}.\mathbbm{1} \, | \, \mathbf{s} \in S(\lambda_1, \lambda_2) \}$$ generates $F_{\lambda_1, \lambda_2}$ as a $U({\mathfrak{n}}^{-})$-modules.\
In our case $\lambda_1 = m_i \omega_i, \lambda_2 =  m_j \omega_j$ and let $\mathbf{s} \in \bz_{\geq 0}^{n(n-1)/2}$ with $s_{p, q} = 0$ for $q < j$ or $p > i$. Let $k, \ell$ be such that $ i-k > \ell - j$, $s_{k, \ell} \neq 0$ and $$\begin{array}{ll}
\text{Condition }(1):&  s_{r, \ell} = 0, \;  \forall \, r=1, \ldots , k-1 \\
\text{Condition }(2): &  s_{r,s} = 0 \text{ if } r < k \text{ and } s< j + i -r, \text{ then } 
\end{array}$$ So $\mathbf{s}$ is of the form: $$\left(
\begin{array}{cccccccccccc}
0 &     \ldots         &      &      0     &     0                &             \ldots                              &      &  &             \ldots         &             0       & s_{i,j}\\
  &           &  &       &                   &            \ldots                               &       &  &       \ldots            &             s_{i-1, j +1}       & s_{i,j+1}\\
\vdots &   \ddots         &     &   \vdots         &        \vdots              &                                &       &           &     \iddots &               & \vdots\\
 0&  &  &   0   &     0           & \ldots  &0  &   s_{i+j - \ell+1, \ell -1}  & \ldots  & & s_{i, \ell-1}\\
0 & \ldots & 0 & s_{k, \ell} & s_{k+1, \ell} & \ldots & s_{i+j - \ell, \ell} & &  \ldots&& s_{i, \ell}\\
s_{1, \ell+1} & \ldots &  &  &  & \ldots & & &  \ldots&& s_{i, \ell+1}\\
\vdots & \ddots &  &  & & \ddots& & &\ddots &&\vdots\\
s_{1, n-1} & \ldots &  & & & \ldots &  & &  \ldots&& s_{i, n-1}
\end{array}
\right)$$ We consider $$f_{k,k} \circ \left(   \mathbf{f}^{\mathbf{s}  - \mathbf{e}_{k, \ell} + \mathbf{e}_{k+1,\ell}}\right).\mathbbm{1}$$ By expanding this we see that $$(s_{k+1, \ell}+1)\mathbf{f}^{\mathbf{s}}. \mathbbm{1} = \left[ f_{k,k} \circ \left(   \mathbf{f}^{\mathbf{s} - \mathbf{e}_{{k, \ell}} + \mathbf{e}_{{k+1,\ell}}}\right) - 
\sum\limits_{z = \ell +1}^{n-1} s_{k+1, z}
 \left(  \mathbf{f}^{\mathbf{s} - \mathbf{e}_{k,\ell}  + \mathbf{e}_{k+1, \ell} + \mathbf{e}_{k,  z} - \mathbf{e}_{k+1, z} } \right)\right]. \mathbbm{1}.$$ By iterating this we see that $$\mathbf{f}^{\mathbf{s}}. \mathbbm{1} \in \sum_{ \mathbf{n}_\alpha} U({\mathfrak{n}}^-) \mathbf{f}^{\mathbf{n}}. \mathbbm{1}$$ where the sum is over all $\mathbf{n}  \in \bz_{\geq 0}^{n(n-1)/2}$ satisfying Condition (1) and (2) and moreover $n_{k,\ell}=0$.\
Using induction along the first row, then along the second row etc, we see that $$\mathbf{f}^{\mathbf{s}}. \mathbbm{1} \in \sum_{ \mathbf{n}} U({\mathfrak{n}}^-) \mathbf{f}^{\mathbf{n}}. \mathbbm{1}$$ where $n_{{k,\ell}} = 0 $ for all $k,\ell$ with $i - k > \ell - j$.\
A similar computation for the roots below the diagonal shows that we can assume also $n_{{ k , \ell}} = 0$ for all $(k, \ell) \neq (i - q, j + q)$ for some $q$. This proves the first part of the lemma. The claim on highest weight points follows now from the definition of $F_{\lambda_1, \lambda_2}$, namely $$(f_{i - q, j+ q} \otimes t)^K. \mathbbm{1} = 0 \text{ for } K \geq \min\{ m_i , m_j \}.$$

The following gives a stricter upper bound for the set of highest weight points.

Let $i\leq j \in I, m_i, m_j \geq 0$, $p = \min \{i-1, n-1-j \}$, then $$S_{hw}(\lambda_1, \lambda_2) \subseteq \{ a_0 \mathbf{e}_{i,j} + a_1 \mathbf{e}_{i-1, j+1} + \ldots + a_p \mathbf{e}_{i-p, j+p} \, | \, \min\{ m_i, m_j \} \geq a_0 \geq a_1 \geq \ldots \geq a_ p \geq 0 \}.$$

We just have to check that these are the only points of the ones described in Lemma \[diagonal\] whose monomials applied on $\mathbbm{1}$ give vectors of dominant weight. For this, the weight of the vector $$(f_{i,j} \otimes t)^{a_0} (f_{i-1, j+1} \otimes t)^{a_1} \cdots (f_{i- p, j+p})^{a_k}. \mathbbm{1}$$ is equal to $$m_i \omega_i + m_j \omega_j + \sum_{q = 0}^{p} a_p (\omega_{i-q} + \omega_{j+q} -  \omega_{i-q-1} - \omega_{j+q+1} ).$$ This is equal to $$(m_i - a_0) \omega_i + (a_0 - a_1) \omega_{i-1} + \ldots + (a_{p-1} - a_p) \omega_p + (m_j - a_0) \omega_j + \ldots + (a_{p-1} - a_p)\omega_{j+p}$$ which is dominant if and only if $a_0 \geq a_1 \geq \ldots \geq a_p$.

Keep the notation from the proof and set $b_i = a_i - a_{i+1} \geq 0$, then the weight of $$(f_{i,j} \otimes t)^{a_0} (f_{i-1, j+1} \otimes t)^{a_1} \cdots (f_{i- p, j+p})^{a_k}. \mathbbm{1}$$ is equal $$m_i\omega_i + m_j \omega_j + \sum_{q = 0}^{\min\{ i-1, n-1 -j\}} b_q(\omega_{i-q} + \omega_{j+q} - \omega_{i} - \omega_{j})$$ with $$\sum_{q = 0}^{\min\{ i-1, n-1 -j\}} b_q = a_0 \leq \min\{m_i, m_j\}.$$ This implies

Let $i, j\in I$, $m_i, m_j \geq 0$, then $$a_{m_i \omega_i, m_j \omega_j}^{\tau} = c_{m_i \omega_i, m_j \omega_j}^{\tau}$$ for all $\tau \in P^+$ and hence $$F_{m_i \omega_i, m_j \omega_j} \cong_{{\mathfrak{sl}}_n \otimes \bc[t]} V(m_i \omega_i)_{c_1} \ast V(m_j \omega_j)_{c_2}$$ for all $c_1 \neq c_2 \in \bc$.

The Pieri rules {#pieri}
===============

In this section we want to compute the ${\mathfrak{sl}}_n$ decomposition on $F_{\lambda, \omega_j}$ and $F_{\lambda, k \omega_1}$. Mainly, we want to identify them with the fusion product of $V(\lambda)$ and $V(\omega_j)$ (resp. $V(k \omega_1)$). As for the Kirillov-Reshetikhin modules we will show that $a_{\lambda, \omega_j}^{\tau} = c_{\lambda, \omega_j}^{\tau}$ for all $\tau$ and similar for $k \omega_1$. Let us start with the latter case.\
On one hand, using again the Young tableaux combinatorics from [@N2002], we see that the highest weight vectors of $V(\lambda) \otimes V(k \omega_1)$ are parameterized by the set $$T_{\lambda, k \omega_1} := \{ ( b_1, \ldots, b_{n} ) \in \mathbb{Z}^{n}_{\geq 0} \, |, \,  b_1 + \ldots + b_{n} =  k \text{ and } b_j \leq m_{j-1} \; \forall \; j = 2, \ldots, n \}$$ where $\lambda = \sum m_i \omega_i$.\
Let $\mathbf{s} \in S_{hw}(\lambda, k\omega_1) \subseteq S(\lambda, k \omega_1)$, then $s_{i,j} = 0$ if $i \neq 1$. So there is no confusion if we write in the following $s_j$ for $s_{1,j}$. We have $s_{1} + \ldots + s_{n-1} \leq k$. Suppose now $s_{j} > m_j$ for some $j > 1$, then by definition of $F_{\lambda, k \omega_1}$ $$f_{j}^{s_{j}}. \mathbbm{1} = 0$$ This implies, recall the notation of Section \[KR\], $$\mathbf{f}^{\mathbf{s} + s_j \mathbf{e}_{1, j-1} - s_j \mathbf{e}_{1, j}}f_{j}^{s_{j}}. \mathbbm{1} = 0$$ Using commutator relations we have for some constants $c_k$ $$\sum_{k = 0}^{s_j}c_k f_{j}^{k}  \mathbf{f}^{\mathbf{s} + k ( \mathbf{e}_{1, j-1} -  \mathbf{e}_{1, j})}. \mathbbm{1} = 0$$ This implies that $$\mathbf{f}^{\mathbf{s}}.\mathbbm{1} \in \sum_{\mathbf{n}} U({\mathfrak{n}}^-) \mathbf{f}^{\mathbf{n}}.\mathbbm{1}$$ for some $\mathbf{n}$ with $n_\ell = s_\ell$ for $\ell > j$ and $ n_j < s_j$. But this is a contradiction to $\mathbf{s}\in S_{hw}(\lambda, k\omega_1)$.\
This implies that if $\mathbf{s} \in S_{hw}(\lambda, k\omega_1)$ we have $$s_{i,j} = 0 \text{ for } i \neq 1 \; , \; s_{j} \leq m_j \; , \;  s_{1} + \ldots + s_{n-1} \leq k.$$ This implies $|S_{hw}(\lambda, k\omega_1) | \leq |T_{\lambda, k \omega_1} |$. Using now Lemma \[whatever\] we have equality here and so

For $\lambda \in P^+$, $k \geq 0$, we have $$a_{\lambda, k \omega_1}^{\tau} = c_{\lambda, k\omega_1}^{\tau} \text{ for all } \tau \in P^+$$ and so for all $c_1 \neq c_2 \in \bc$ $$F_{\lambda, k \omega_1} \cong V(\lambda)_{c_1}\ast V(k\omega_1)_{c_2}.$$

We consider here the $\omega_j$-case. As before, using Young Tableaux combinatorics from [@N2002], we have that the highest weight vectors of $V(\lambda) \otimes V(\omega_j)$ are parameterized by the set ($\lambda = \sum m_i \omega_i$) $$T_{\lambda, \omega_j} := \{ ( b_1 < \ldots <  b_{j} )   \, |, \, b_i \in \{1, \ldots, n\} \text{ s.t.: }   b_{i-1} \neq b_{i} - 1 \Rightarrow m_{b_i-1} \neq 0\}.$$ Let $\mathbf{s} \in S_{hw}(\lambda, \omega_j) \subseteq S(\lambda, \omega_j)$, then $s_{k, \ell} = 0$ if $\ell > j $ or $k < j$. We have for all Dyck path $\mathbf{p}$: $\beta_1 + \ldots + \beta_s \leq 1$. This implies that $s_\beta \in \{ 0, 1\}$ for all $\beta$ and even more, that the support of $\mathbf{s}_\alpha$ is of the form $$\{ \alpha_{i_1, j_1} , \ldots, \alpha_{i_\ell, j_\ell} \, | \, i_1 < i_2 \ldots < i_\ell \leq j \leq j_\ell < \ldots < j_1 \}$$ Let us parametrize this set as follows. Let $ \alpha_{i_1, j_1} , \ldots, \alpha_{i_\ell, j_\ell}$ be given from the set and denote $$\{ p_1 < \ldots < p_{j-\ell} := \{1, \ldots, j\}\setminus \{ i_1, \ldots, i_\ell \} .$$ Then we associate $$\alpha_{i_1, j_1} , \ldots, \alpha_{i_\ell, j_\ell} \leftrightarrow (p_1 < p_2 < \ldots < p_{j-\ell} < j_{\ell} + 1 < \ldots < j_{1} + 1)$$ This gives a one to one correspondence to $j$-tuples of strictly increasing integers smaller equals to $n$, hence parameterizes a basis of $V(\omega_k)$.\
Since we are interested in the highest weight vectors, we can exclude these tuples corresponding to vectors in $F_{\lambda, \omega_j}$ of non-dominant weight. The weight of such a vector $(p_1 < p_2 < \ldots < p_j)$ is given by $$\lambda + ( \omega_j - ( - \omega_{p_1-1} + \omega_{p_1 } -  \omega_{p_2-1} + \omega_{p_2} -  \ldots - \omega_{p_{j-1}-1} + \omega_{p_{j-1}} \omega_{p_j-1} + \omega_{p_j}).$$ With a short calculation one sees that this is dominant if and only if $p_i \neq p_{i+1} - 1 \Rightarrow m_{p_i}-1 > 0$.\
This implies that $a_{\lambda, \omega_j}^{\tau} \leq c_{\lambda, \omega_j}^{\tau}$ for all $\tau \in P^+$. Using Lemma \[whatever\] implies now equality for all $\tau$ which proves

For $\lambda \in P^+$, $j \in \{ 1, \ldots, n-1\} = I$, we have $$a_{\lambda, \omega_j}^{\tau} = c_{\lambda, \omega_j}^{\tau} \text{ for all } \tau \in P^+$$ and so for all $c_1 \neq c_2 \in \bc$ $$F_{\lambda, \omega_j} \cong V(\lambda)_{c_1}\ast V(\omega_j)_{c_2}.$$

Partial order and Weyl modules {#poset}
==============================

In [@CFS14] a partial order on pairs of dominant weight has been introduced. Let us recall here briefly the construction. Fix $\lambda \in P^+$ and consider the partitions of $\lambda$ with two parts $$P(\lambda,2) = \{ (\lambda_1, \lambda_2) \in P^+ \times P^+ \, | \, \lambda_1 + \lambda_2 = \lambda \}.$$ By abuse of notation we denote by $P(\lambda,2)$ the orbits of the natural $S_2$ action on $P(\lambda,2)$. In [@CFS14], the following partial order has been introduced on $P(\lambda,2)$: Let $\blambda = ( \lambda_1, \lambda - \lambda_1), \bmu = (\mu_1, \lambda - \mu_1) \in P(\lambda,2)$, then $$\blambda \preceq \bmu : \Leftrightarrow \; \forall \, \alpha \in R^+ : \min\{ \lambda_1(h_\alpha), (\lambda - \lambda_1)(h_\alpha)\} \leq \min \{ \mu_1(h_\alpha), (\mu - \mu_1)(h_\alpha) \}.$$ Certain properties of this poset were proved in [@CFS14] (and [@F14b]), e.g. there exists a smallest element in $P(\lambda,2)$, the orbit of $(\lambda, 0)$. It is less obvious that there exists also a unique maximal element: let $\lambda = \sum_{i = 1}^{n-1} m_i \omega_i$, and let $\{1\leq i_1 < \ldots <  i_k \} =  I_{odd}$ be the indices such that $m_i$ is odd. Then $\blambda^{\max}  = (\lambda_1^{\max}, \lambda_2^{\max})$ given by $$\lambda_1^{\max}=
  \sum_{j=1}^k ((m_{i_j}+ (-1)^j)/2)\omega_{i_s}+\sum_{i\in I \setminus I_{odd}}(m_i/2)\omega_i, \qquad
  \lambda_2^{\max}=\lambda-\lambda_1^{\max},$$ is the unique maximal orbit in $P(\lambda,2)$, [@CFS14 Proposition 5.3].\
It was further shown that the cover relation of $\preceq$ on $P(\lambda,2)$ is determined by the Weyl group action [@CFS14 Proposition 6.1].

We want to relate the partial order and the modules $F_{\lambda_1, \lambda_2}$. Namely, we want to prove the following lemma:

\[fusion-sp\] Suppose $(\lambda_1, \lambda - \lambda_1) \preceq (\mu_1, \lambda - \mu_1) \in P(\lambda,2)$, then there exists a canonical surjective map of ${\mathfrak{sl}}_n \otimes \bc[t]$-modules $$F_{\mu_1, \mu - \mu_1} \twoheadrightarrow F_{\lambda_1, \lambda - \lambda_1}.$$

We have to compare the defining relations only. So let $\alpha \in R^+$, then on both modules we have $$(f_\alpha \otimes 1)^{\lambda(h_\alpha) + 1}.\mathbbm{1} = 0$$ and also the highest weight is in both cases $\lambda$. Let $M_1 = \min\{ \mu_1(h_\alpha), \lambda - \mu_1(h_\alpha) \}$ and $M_2 = \min\{ \lambda_1(h_\alpha), \lambda - \lambda_1(h_\alpha) \}$, then by assumption $M_1 \geq M_2$. By the defining relations of $F_{\lambda_1, \lambda - \lambda_1}$ we have $$(f \otimes t)^{M_2 + 1}. \mathbbm{1} = 0 \in F_{\lambda_1, \lambda - \lambda_1}$$ so especially $$(f \otimes t)^{M_1 + 1}. \mathbbm{1} = 0 \in F_{\lambda_1, \lambda - \lambda_1}.$$ This implies the lemma.

We turn to the unique maximal element in $P(\lambda,2)$, $\blambda^{\max} =  (\lambda_1^{\max}, \lambda_2^{\max})$. In fact we want to identify $F_{ \lambda_1^{\max}, \lambda_2^{\max}}$ as the unique graded local Weyl module of ${\mathfrak{sl}}_n \otimes \bc[t]/(t^2)$ of highest weight $\lambda$. For this we recall the definition of a local Weyl module briefly in the following.\
Let $A$ be a commutative, finitely generated unital algebra over $\bc$. Then ${\mathfrak{sl}}_n \otimes A$ is a Lie algebra with bracket given by $$[x \otimes p, y \otimes q]= [x,y] \otimes pq$$ and it is called the generalized current algebra. We fix $\lambda \in P^+$, this induces an one-dimensional ${\mathfrak{h}}$-modules, which we denote $\bc_{\lambda}$. Let $\xi: ({\mathfrak{n^+}} \oplus {\mathfrak{h}}) \otimes A \longrightarrow {\mathfrak{h}}$ be a Lie algebra homomorphism. Then we can lift the structure on $\bc_{\lambda}$ to a $ ({\mathfrak{n^+}} \oplus {\mathfrak{h}}) \otimes A$-structure, and let us denote this one-dimensional module $\bc_{\lambda, \xi}$.

The local Weyl module $W_A(\xi, \lambda)$ is unique maximal integrable (as a ${\mathfrak{sl}}_n$-module) quotient of the ${\mathfrak{sl}}_n \otimes A$-module $$U({\mathfrak{sl}}_n \otimes A) \otimes_{ ({\mathfrak{n}}^+ \oplus {\mathfrak{h}} ) \otimes A} \bc_{\lambda, \xi}.$$

These modules have been introduced for $A=  \bc[t^{\pm 1}]$ in [@CP01] and further generalized in [@FL04] and [@CFK10] to arbitrary commutative associative algebras over $\bc$. It has been shown in [@CFK10] that if $A$ is finitely generated, $W_A(\xi, \lambda)$ is finite-dimensional and further that these modules are parameterized by maximal ideals in a tensor product of symmetric powers of $A$.\
These modules play an important role in the representation theory of ${\mathfrak{sl}}_n \otimes A$, the interested reader is here referred to [@CFK10].\
As they are integrable as ${\mathfrak{sl}}_n$-modules, there exist a decompositions into finite-dimensional simple ${\mathfrak{sl}}_n$-modules. Unfortunately, these decomposition are known for special cases only. Namely for $A= \bc[t], \bc[t^{\pm 1}]$ they are computed in a series on paper [@CP01], [@CL06], [@FoL07]. If $A$ is semi-simple, then the local Weyl module obviously decomposes into a direct sum of local Weyl modules for ${\mathfrak{sl}}_n \otimes \bc = {\mathfrak{sl}}_n$, so into a direct sum of simple ${\mathfrak{sl}}_n$-modules.\
But outside of these cases, even for the “smallest“ non-semi-simple algebra $A = \bc[t]/(t^2)$, the ${\mathfrak{sl}}_n$ decomposition is unknown.\
Let us rewrite the defining relations for the local Weyl modules for $A =  \bc[t]/(t^K)$. In fact, for each $\lambda \in P^+$ and $K\geq 1$, there exists a unique local Weyl module. This follows since there exists a unique non-trivial map $\lambda \circ \xi$, namely $\xi$ is the evaluation map at $t = 0$, so we denote $\xi$ by $0$.

\[localweyl\] Let $\lambda \in P^+$, then the *graded local Weyl module* $W_{\bc[t]/(t^K)}(0,\lambda)$ is generated by $w \neq 0$ with relations $$({\mathfrak{n}}^+ \oplus {\mathfrak{h}}) \otimes t.w = 0 \;, \;  h - \lambda(h). w = 0  \; , \; {\mathfrak{n}}^+.w = 0 \; , \; (f_{\alpha} \otimes 1)^{\lambda(h_\alpha) + 1}.w = 0 .$$

Since the relations are homogeneous, we see that $W_{\bc[t]/(t^K)}(0,\lambda)$ is a graded ${\mathfrak{sl}}_n \otimes \bc[t]/(t^K)$-module. Even more, we have immediately from the defining relations

\[f-weyl\] Let $\lambda_1 + \lambda _2 = \lambda \in P^+$ and $K \geq 2$, then there exists a surjective map of ${\mathfrak{sl}}_n \otimes \bc[t]$-modules $$W_{\bc[t]/(t^K)}(0,\lambda) \twoheadrightarrow F_{\lambda_1, \lambda_2}.$$

In fact $ F_{\lambda_1, \lambda_2}$ is the quotient obtained by factorizing the $U( {\mathfrak{sl}}_n \otimes \bc[t])$-submodule generated by $$\{ (f_\alpha \otimes t)^{\min\{ \lambda_1(h_\alpha), \lambda_2(h_\alpha) \} +1 }.\mathbbm{1} \, | \, \alpha \in R^+ \} \cup \{ f_\alpha \otimes t^{\ell} \, | \, \ell \geq 2, \alpha \in R^+ \}.$$

In this subsection we are restricting ourselves to the case of the second truncated current algebra, and we denote $A = \bc[t]/(t^2)$. We will prove

\[weyl-f\] Let $\lambda \in P^+$ and $\blambda^{\max} =  (\lambda_1^{\max}, \lambda_2^{\max})$ be the unique maximal element in $P(\lambda, 2)$. Then we have an isomorphism of ${\mathfrak{sl}}_n \otimes A$-modules (and by extending an isomorphism of ${\mathfrak{sl}}_n \otimes \bc[t]$-modules): $$W_A(0,\lambda) \cong F_{ \lambda_1^{\max}, \lambda_2^{\max}}.$$

We consider the ${\mathfrak{sl}}_2$-case first. Then $\lambda = m \omega$ and because $e, e \otimes t, h \otimes t$ are acting trivial on $\mathbbm{1}$, $$W_A(0,\lambda) = \operatorname{span} \{ f^K (f \otimes t)^L .w \,| ,\ K, L \geq 0 \}$$ So if we restrict to elements in degree $L$ (recall, that $W_A(0,\lambda)$ is graded by the degree of $t$), then this is spanned by $$\{  f^K (f \otimes t)^L.w \, | \, K \geq 0 \}$$ The weights in degree $L$ are therefore of the form $m - 2L - 2K$ with $K \geq 0$. Every graded component is a finite-dimensional ${\mathfrak{sl}}_2$-module, since ${\mathfrak{sl}}_2$ acts by degree $0$ and $W_A(0,\lambda)$ is finite-dimensional. This implies that the component of degree $L$ in $W(0,\lambda)$ is $0$ if there is no vector of dominant weight in degree $L$. So for $L > \lfloor m/2 \rfloor$ we have $(f  \otimes t)^L.w = 0$.\
On the other hand, $\blambda = ( \lfloor m/2 \rfloor, \lceil m/2 \rceil )$ which implies that $ F_{\lambda_1^{\max}, \lambda_2^{\max} }$ is the quotient by the submodule generated by $$(f \otimes t)^L.\mathbbm{1}  \text{ with } L > \lfloor m/2 \rfloor.$$ This implies that $W_A(0,\lambda) \cong F_{\lambda_1^{\max}, \lambda_2^{\max} }$.\
Let us turn to the general case. We have $$\min \{ \lambda_1^{\max}(h_\alpha), \lambda_2^{\max}(h_\alpha) \}  = \lfloor \lambda(h_\alpha)/2 \rfloor.$$ It is enough to show that $(f_\alpha \otimes t)^{ \lfloor \lambda(h_\alpha)/2 \rfloor + 1}.\mathbbm{1} = 0 \in W_A(0, \lambda)$ for all $\alpha$.\
Fix $\alpha > 0$ and consider the Lie subalgebra ${\mathfrak{sl}}(\alpha) \otimes A =  \langle e_\alpha, h_\alpha, f_\alpha, e_\alpha \otimes t, h_\alpha \otimes t, f_\alpha \otimes t \rangle$ which is isomorphic to ${\mathfrak{sl}}_2 \otimes A$.\
We consider the submodule $M = U({\mathfrak{sl}}(\alpha) \otimes A).\mathbbm{1} \subseteq W_A(0,\lambda)$. Then this is a quotient of the ${\mathfrak{sl}}_2 \otimes A$ local Weyl module $W_A(0, \lambda(h_\alpha) \omega)$ (since the defining relations are satisfied on the highest weight vector).\
The considerations above for the ${\mathfrak{sl}}_2$-case imply now that $$(f_\alpha \otimes t)^{\lfloor \lambda(h_\alpha)/2 \rfloor + 1}.\mathbbm{1} = 0  \in M \subseteq W_A(0,\lambda)$$ Which implies that $W_A(0, \lambda)$ is a quotient of $F_{\lambda_1^{\max}, \lambda_2^{\max}}$ and hence they are isomorphic.

[^1]: The author was partially supported by the DFG priority program 1388 “Representation Theory“
