Kronecker coefficients for some near-rectangular partitions

Abstract

Abstract. We give formulae for computing Kronecker coefficients occurring in the expansion of sµ * sν, where both µ and ν are nearly rectangular, and have smallest parts equal to either 1 or 2. In particular, we study s (n,n−1,1) * s (n,n) , s (n−1,n−1,1) * s (n,n−1) , s (n−1,n−1,2) * s (n,n) , s (n−1,n−1,1,1) * s (n,n) and s (n,n,1) * s (n,n,1) . Our approach relies on the interplay between manipulation of symmetric functions and the representation theory of the symmetric group, mainly employing the Pieri rule and a useful identity of Littlewood. As a consequence of these formulae, we also derive an expression enumerating certain standard Young tableaux of bounded height, in terms of the Motzkin and Catalan numbers.An outstanding open problem in algebraic combinatorics is to derive a combinatorial formula to compute the Kronecker product of two Schur functions. Given partitions λ, µ and ν, the Kronecker coefficients, g λ µν , occur in the decomposition of the Kronecker product s µ * s ν of Schur functions in the Schur basis.Alternatively, these coefficients can also be defined as the multiplicities of the irreducible representations of the symmetric group in the tensor product of two irreducible representations of the symmetric group. This interpretation immediately implies that the Kronecker coefficients are non-negative integers leading one to believe that there should be a combinatorial rule to compute these coefficients. However, to date, there is no satisfactory positive combinatorial formula for the Kronecker product of two Schur functions.Besides the intrinsic interest in the problem, the motivation for discovering a combinatorial formula is the impact beyond algebraic combinatorics. For example, the Kronecker coefficients arise in quantum information theory and quantum computation [13,14,34]. The problem of computing them combinatorially has got major impetus from the fact that they are of prime importance in Geometric Complexity Theory, a program of Mulmuley aimed at resolving the P vs NP problem [24]. In other applications, these coefficients have been used to show the strict unimodality of q-binomial numbers by Pak and Panova [25].Attempts have been made to understand different aspects of these coefficients, for example, special cases [5,6,10,27,28,33], asymptotics [1,2], stability [9,32], the complexity of computing them and conditions which guarantee that they are non-zero [12]. Recently, a combinatorial rule was given by Blasiak [7] for computing s µ * s ν where at least one of µ and ν is a hook shape. Finally, a certain variant, called the reduced Kronecker coefficients, has also been studied in [8,9].

Rectangular Kronecker coefficients and plethysms in geometric complexity theory

“…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. As shown in [IMW17], computing the Kronecker coefficients is #P-hard, and deciding positivity is NP-hard.…”

Maximal lexicographic spectra and ranks for states with fixed uniform margins

“…On the other hand, let γ = (4 k−1 , 3, 3). By Theorem 1.6 of [18] and Proposition 3.1, we have that g(λ, µ; γ) = g(λ, µ t ; γ t ) = 1. Hence, γ = (4 k−1 , 3, 3) ∈ Φ(λ, µ).…”

A virtual character and nonzero Kronecker coefficients for self-conjugate partitions

“…Generally, the evaluation is complicated. But in some special cases this formula is practicable, for example [20]. In this section we will provide another example.…”