All Kronecker coefficients are reduced Kronecker coefficients

We settle the question of where exactly do the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a reduced Kronecker coefficient by an explicit construction. T...

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Bibliographic Details
Published in:Forum of mathematics. Pi 2024-01, Vol.12, Article e22
Main Authors: Ikenmeyer, Christian, Panova, Greta
Format: Article
Language:English
Subjects:
05E05
20C15
20C30
20G05
22E46
Combinatorial analysis
Combinatorics
Discrete Mathematics
Isomorphism
Vector spaces
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Summary:We settle the question of where exactly do the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a reduced Kronecker coefficient by an explicit construction. This implies the equivalence of an open problem by Stanley from 2000 and an open problem by Kirillov from 2004 about combinatorial interpretations of these two families of coefficients. Moreover, as a corollary, we deduce that deciding the positivity of reduced Kronecker coefficients is ${\textsf {NP}}$ -hard, and computing them is ${{{\textsf {#P}}}$ -hard under parsimonious many-one reductions. Our proof also provides an explicit isomorphism of the corresponding highest weight vector spaces.
ISSN:2050-5086
2050-5086
DOI:10.1017/fmp.2024.23