Multiplicity-free Kronecker products of characters of the symmetric groups

Abstract

Abstract. We provide a classification of multiplicity-free inner tensor products of irreducible characters of symmetric groups, thus confirming a conjecture of Bessenrodt. Concurrently, we classify all multiplicity-free inner tensor products of skew characters of the symmetric groups. We also provide formulae for calculating the decomposition of these tensor products.

Splitting Kronecker squares, 2-decomposition numbers, Catalan Combinatorics, and the Saxl conjecture

“…We first state the decomposition of some multiplicity-free products of characters to 2-part partitions. These follow from the general formulae, and were known already as part of the easy direction of the classification conjecture on multiplicity-free products (see [5]); they have also appeared explicitly in some more recent papers. The fact that only partitions of length at most 4 can appear is easy to see as a consequence of [13].…”

“…The bulk of the paper is dedicated to advancing our understanding of symmetric and anti-symmetric Kronecker coefficients by analogy with well-known milestones in the classical theory Kronecker coefficients. These milestones include: the classification of homogeneous and irreducible products [2]; the classification of multiplicity-free Kronecker products [5]; partial and complete results for special classes of partitions (such as hooks [6,25], 2-line partitions [1,39], partitions of small depth [40,44,48], and rectangles [27,28]); and most recently, Saxl's Kronecker positivity conjecture. We provide the analogue of the Bessenrodt-Kleshchev classification of multiplicity-free products for the symmetric and anti-symmetric Kronecker squares.…”

Kronecker positivity and 2-modular representation theory

“…classified in [BB17]. For constituents to partitions of depth at most 4, explicit formulae for their multiplicity in squares were provided by Saxl in 1987, and later work by Zisser and Vallejo, respectively.…”

Kronecker positivity and 2-modular representation theory

“…We wish to provide bounds on the Kronecker coefficients: the maximal possible values obtained by Kronecker products are studied in [PPV16], and the Kronecker products whose coefficients are all as small as possible (namely all 0 or 1) are classified in [BB17]. For constituents to partitions of depth at most 4, explicit formulae for their multiplicity in squares were provided by Saxl in 1987, and later work by Zisser and Vallejo, respectively.…”