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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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Specht modules decompose as alternating sums of restrictions of Schur modules
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by Sami H. Assaf and David E. Speyer PDF
Proc. Amer. Math. Soc. 148 (2020), 1015-1029 Request permission


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
  • MR Author ID: 775302
  • Email:
  • David E. Speyer
  • Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 28109-1043
  • MR Author ID: 663211
  • Email:
  • 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:
  • MathSciNet review: 4055931