Each affine Coxeter group is associated to an affine Lie algebra, a certain infinite-dimensional non-associative algebra with unusually nice representation-theoretic properties. In this association, the Coxeter group arises as a group of symmetries of the root space of the Lie algebra (the dual of the Cartan subalgebra). In the classification of affine Lie algebras, the one associated to S ~ ~ n is of (untwisted) type A n − − 1 ( 1 ) , with Cartan matrix [ 2 − − 2 − − 2 2 ] for n = 2 and [ 2 − − 1 0 ⋯ ⋯ 0 − − 1 − − 1 2 − − 1 ⋯ ⋯ 0 0 0 − − 1 2 ⋯ ⋯ 0 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋮ ⋮ 0 0 0 ⋯ ⋯ 2 − − 1 − − 1 0 0 ⋯ ⋯ − − 1 2 ] (a circulant matrix) for n > 2 .