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Specht modules decompose as alternating sums of restrictions of Schur modules


Authors: Sami H. Assaf and David E. Speyer
Journal: Proc. Amer. Math. Soc. 148 (2020), 1015-1029
MSC (2010): Primary 20C15; Secondary 20C30, 05E05, 05E10
DOI: https://doi.org/10.1090/proc/14815
Published electronically: October 28, 2019
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Abstract: Schur modules give the irreducible polynomial representations of the general linear group $ \mathrm {GL}_t$. Viewing the symmetric group $ \mathfrak{S}_t$ as a subgroup of $ \mathrm {GL}_t$, we may restrict Schur modules to $ \mathfrak{S}_t$ and decompose the result into a direct sum of Specht modules, the irreducible representations of $ \mathfrak{S}_t$. We give an equivariant Möbius inversion formula that we use to invert this expansion in the representation ring for $ \mathfrak{S}_t$ for $ t$ large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternate in sign by degree. In particular, this allows us to define a new basis of symmetric functions whose structure constants are stable Kronecker coefficients and which expand with signs alternating by degree into the Schur basis.


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Additional Information

Sami H. Assaf
Affiliation: Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, California 90089-2532
Email: shassaf@usc.edu

David E. Speyer
Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 28109-1043
Email: speyer@umich.edu

DOI: https://doi.org/10.1090/proc/14815
Received by editor(s): October 14, 2018
Received by editor(s) in revised form: July 12, 2019
Published electronically: October 28, 2019
Additional Notes: The first author was supported by NSF DMS-1763336
The second author was supported by NSF DMS-1600223
Communicated by: Benjamin Brubaker
Article copyright: © Copyright 2019 American Mathematical Society