Kronecker coefficients and noncommutative super Schur functions

Abstract

The theory of noncommutative Schur functions can be used to obtain positive combinatorial formulae for the Schur expansion of various classes of symmetric functions, as shown by Fomin and Greene [11]. We develop a theory of noncommutative super Schur functions and use it to prove a positive combinatorial rule for the Kronecker coefficients g λµν where one of the partitions is a hook, recovering previous results of the two authors [5,21]. This method also gives a precise connection between this rule and a heuristic for Kronecker coefficients first investigated by Lascoux [19].• reprove and strengthen the rule from [21],• establish a precise connection between this rule and the Lascoux heuristic, and • uncover a surprising parallel between this rule and combinatorics underlying transformed Macdonald polynomials indexed by a 3-column shape, as described in [14,6].

Rectangular Kronecker coefficients and plethysms in geometric complexity theory

“…In 2006 Ballantine and Orellana [BO06] established a rule for k(λ, µ, ν) when µ = (n − k, k) and λ 1 ≥ 2k − 1. The most general rule for ν = (n − k, 1 k ), a hook, and any other two partitions, was estabslished by Blasiak in 2012 [Bla17], and later simplified in [Liu17,BL18]. Other special cases include multiplicity-free Kronecker products by Bessenrodt-Bowman [BB17], triples of partitions which are marginals of pyramids by Ikenmeyer-Mulmuley-Walter [IMW17], k(m k , m k , (mk − n, n)) as counting labeled trees by Pak-Panova [Pan15, slide 9], near-rectangular partitions by Tewari in [Tew15], etc.…”

Noncommutative Schur functions for posets

“…, 6}, the chains a = (6, 4, 3, 2) and b = (5, 4, 1) produce a pairing of 4 with 4, 5 with 3, 1 with 6. Then after transferring 2 we get chains (6,4,3) and (5, 4, 2, 1). The associated word of parentheses is )((())( and after transferring 2 it becomes ))(())(.…”

“…Drawing on ideas of Lascoux and Schützenberger [30,34], Fomin and Greene [15] developed this theory to give positive formulae for the Schur expansions of a large class of symmetric functions that includes the Stanley symmetric functions and stable Grothendieck polynomials. More recently, Fomin, Liu, and the first author [4,5,6] further developed this theory, incorporating ideas of Assaf [2,3] and Lam [29], and used it to prove Schur positivity of LLT polynomials indexed by a 3-tuple of skew shapes and to give a new proof of a formula for Kronecker coefficients when one of the shapes is a hook.…”

Rules of Three for commutation relations

“…The ring defined by the relations (1.14)-(1.15) has many quotients with rich combinatorial structure (the plactic algebra, nilCoxeter algebra, and more, see [4,6,10,13]); the ring defined by the relations (1.18)-(1.19) has many interesting quotients as well, some of them similar to the plactic algebra. The recent paper [7] studies some of these quotients and develops an accompanying theory of noncommutative super Schur functions. The main application (recovering results of [3,16]) is a positive combinatorial rule for the Kronecker coefficients where one of the shapes is a hook.…”