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Restriction of odd degree characters and natural correspondences
Let be an odd prime power, , and let denote a maximal
parabolic subgroup of with Levi subgroup . We restrict the odd-degree irreducible characters of to
to discover a natural correspondence of characters, both for and
. A similar result is established for certain finite groups with
self-normalizing Sylow -subgroups. We also construct a canonical bijection
between the odd-degree irreducible characters of and those of , where
is any maximal subgroup of of odd index; as well as between the
odd-degree irreducible characters of or with odd
and those of , where is a Sylow -subgroup of . Since our
bijections commute with the action of the absolute Galois group over the
rationals, we conclude that the fields of values of character correspondents
are the same. We use this to answer some questions of R. Gow
Kronecker products and the RSK correspondence
The starting point for this work is an identity that relates the number of
minimal matrices with prescribed 1-marginals and coefficient sequence to a
linear combination of Kronecker coefficients. In this paper we provide a
bijection that realizes combinatorially this identity. As a consequence we
obtain an algorithm that to each minimal matrix associates a minimal component,
with respect to the dominance order, in a Kronecker product, and a
combinatorial description of the corresponding Kronecker coefficient in terms
of minimal matrices and tableau insertion. Our bijection follows from a
generalization of the dual RSK correspondence to 3-dimensional binary matrices,
which we state and prove. With the same tools we also obtain a generalization
of the RSK correspondence to 3-dimensional integer matrices
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