Rectangular Schur functions and the basic representation of affine Lie algebras

Abstract

An expression is given for the plethysm p 2 • S , where p 2 is the power sum of degree two and S is the Schur function indexed by a rectangular partition. The formula can be well understood from the viewpoint of the basic representation of the affine Lie algebra of type A (2) 2 .

Bounds on certain classes of Kronecker and q-binomial coefficients

“…This simple but very useful lemma was first proved in a special case in [CM], and in full generality in [Val,§7] and later in [PP1], but is implicit in [MY,PP2]. Note that it immediately implies Sylvester's unimodality theorem.…”

Strict unimodality of q-binomial coefficients

“…For 3 < k ≤ ⌊n/2⌋ − 1 we have that k ≤ ⌊ℓm 1 /2⌋ + ⌊ℓm 2 /2⌋, so there are values r, s ≥ 2, r ≤ ⌊ℓm 1 /2⌋ and s ≤ ⌊ℓm 2 /2⌋, such that k = r + s. Finally, when k = ⌊n/2⌋, by the parity conditions we have that at least one of ℓm 1 , ℓm 2 is even, so we can choose (r, s) = (ℓm 1 /2, ⌊ℓm 2 /2⌋) or (⌊ℓm 1 /2⌋, ℓm 2 /2). {(5, 6), (5,10), (5,14), (6,6), (6,7), (6,9), (6,11), (6,13), (7, 10)}. Now, case ℓ = 2 is straightforward, since p 2i (2, m) = p 2i+1 (2, m) for all i < n/4.…”

“…= 1 if α and β are complementary partitions within the rectangle (m ℓ ); and c (m ℓ ) αβ = 0 otherwise (see e.g. [10]). Therefore,…”

Mixed expansion formula for the rectangular Schur functions and the affine Lie algebra A1(1)

“…Let (m, n) denote the Young diagram of the rectangular shape (n m ). Set also S μ (t (2) ) = S μ (t 2 , t 4 , t 6 , . .…”

“…The rectangular S-functions are studied in [3,6] from a viewpoint of representations of the affine Lie algebra of type A (1) 1 and A (2) 2 . These functions appear as certain distinguished weight vectors in the so called homogeneous realization of the basic representation L(Λ 0 ) of A (1) 1 (see [5]).…”