99 research outputs found
The partition algebra and the plethysm coefficients
We propose a new approach to study plethysm coefficients by using the
Schur-Weyl duality between the symmetric group and the partition algebra. This
allows us to explain the stability properties of plethysm and Kronecker
coefficients in a simple and uniform fashion for the first time. We prove the
strengthened Foulkes' conjecture for stable plethysm coefficients in an
elementary fashion
Plethysm and lattice point counting
We apply lattice point counting methods to compute the multiplicities in the
plethysm of . Our approach gives insight into the asymptotic growth of
the plethysm and makes the problem amenable to computer algebra. We prove an
old conjecture of Howe on the leading term of plethysm. For any partition
of 3,4, or 5 we obtain an explicit formula in and for the
multiplicity of in .Comment: 25 pages including appendix, 1 figure, computational results and code
available at http://thomas-kahle.de/plethysm.html, v2: various improvements,
v3: final version appeared in JFoC
Brill-Gordan Loci, Transvectants and an Analogue of the Foulkes Conjecture
Combining a selection of tools from modern algebraic geometry, representation
theory, the classical invariant theory of binary forms, together with explicit
calculations with hypergeometric series and Feynman diagrams, we obtain the
following interrelated results. A Castelnuovo-Mumford regularity bound and a
projective normality result for the locus of hypersufaces that are equally
supported on two hyperplanes. The surjectivity of an equivariant map between
two plethystic compositions of symmetric powers; a statement which is
reminiscent of the Foulkes-Howe conjecture. The nonvanishing of even
transvectants of exact powers of generic binary forms. The nonvanishing of a
collection of symmetric functions defined by sums over magic squares and
transportation matrices with nonnegative integer entries. An explicit set of
generators, in degree three, for the ideal of the coincident root locus of
binary forms with only two roots of equal multiplicity.Comment: This is a considerably expanded version of math.AG/040523
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Mini-Workshop: Kronecker, Plethysm, and Sylow Branching Coefficients and their Applications to Complexity Theory
The Kronecker, plethysm and Sylow branching coefficients describe the decomposition of representations of symmetric groups obtained by tensor products and induction.
Understanding these decompositions has been hailed as
one of the definitive open problems in algebraic combinatorics and has profound and deep connections with representation theory, symplectic geometry, complexity theory, quantum information theory, and local-global conjectures in representation theory of finite groups.
The overarching theme of the Mini-Workshop has been the use of hidden, richer representation theoretic structures to
prove and disprove conjectures concerning these coefficients.
These structures arise from
the
modular and local-global representation theory of symmetric groups,
graded representation theory of Hecke and Cherednik algebras, and categorical Lie theory
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