Specht modules decompose as alternating sums of restrictions of Schur modules
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- by Sami H. Assaf and David E. Speyer
- Proc. Amer. Math. Soc. 148 (2020), 1015-1029
- 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.References
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Bibliographic Information
- Sami H. Assaf
- Affiliation: Department of Mathematics, University of Southern California, 3620 S. Vermont Avenue, Los Angeles, California 90089-2532
- MR Author ID: 775302
- Email: shassaf@usc.edu
- David E. Speyer
- Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 28109-1043
- MR Author ID: 663211
- Email: speyer@umich.edu
- 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
- © Copyright 2019 American Mathematical Society
- 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
- MathSciNet review: 4055931