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
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
- Anders Björner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), no. 1, 159–183. MR 570784, DOI 10.1090/S0002-9947-1980-0570784-2
- Michel Brion, Stable properties of plethysm: on two conjectures of Foulkes, Manuscripta Math. 80 (1993), no. 4, 347–371. MR 1243152, DOI 10.1007/BF03026558
- C. C. Cadogan, The Möbius function and connected graphs, J. Combinatorial Theory Ser. B 11 (1971), 193–200. MR 284377, DOI 10.1016/0095-8956(71)90029-3
- Philip Hall, The collected works of Philip Hall, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1988. Compiled and with a preface by K. W. Gruenberg and J. E. Roseblade; With an obituary by Roseblade. MR 986732
- D. E. Littlewood, Group Characters and the Structure of Groups, Proc. London Math. Soc. (2) 39 (1935), no. 2, 150–199. MR 1576896, DOI 10.1112/plms/s2-39.1.150
- D. E. Littlewood, Products and plethysms of characters with orthogonal, symplectic and symmetric groups, Canadian J. Math. 10 (1958), 17–32. MR 95209, DOI 10.4153/CJM-1958-002-7
- D. E. Littlewood and A. R. Richardson, Group characters and algebra, Philos. Trans. Roy. Soc. London. Ser. A. 233 (1934), 99–141.
- Dudley E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, Oxford University Press, New York, 1940. MR 0002127
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
- F. D. Murnaghan, The Analysis of the Kronecker Product of Irreducible Representations of the Symmetric Group, Amer. J. Math. 60 (1938), no. 3, 761–784. MR 1507347, DOI 10.2307/2371610
- Rosa Orellana and Mike Zabrocki, Symmetric group characters as symmetric functions, arXiv:1605.06672, 2016.
- Rosa Orellana and Mike Zabrocki, Products of characters of the symmetric group, arXiv:1709.08098, 2017.
- Mario Pieri, Sul problema degli spazi secanti, Rend. Ist. Lombardo (2) 26 (1893), 534–546.
- Mohan S. Putcha, Complex representations of finite monoids, Proc. London Math. Soc. (3) 73 (1996), no. 3, 623–641. MR 1407463, DOI 10.1112/plms/s3-73.3.623
- Eric M. Rains, The action of $S_n$ on the cohomology of $\overline M_{0,n}(\Bbb R)$, Selecta Math. (N.S.) 15 (2009), no. 1, 171–188. MR 2511203, DOI 10.1007/s00029-008-0467-8
- G. de B. Robinson, On the Representations of the Symmetric Group, Amer. J. Math. 60 (1938), no. 3, 745–760. MR 1507943, DOI 10.2307/2371609
- Christopher Ryba, Resolving irreducible $\mathbb {C}S_n$-modules by modules restricted from $GL_n(\mathbb {C})$, arXiv:1812.07212, 2018.
- M.-P. Schützenberger, La correspondance de Robinson, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976), Springer, Berlin, 1977, pp. 59–113. Lecture Notes in Math., Vol. 579.
- Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
- Sheila Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. Math. 104 (1994), no. 2, 225–296. MR 1273390, DOI 10.1006/aima.1994.1030
- Sheila Sundaram and Volkmar Welker, Group representations on the homology of products of posets, J. Combin. Theory Ser. A 73 (1996), no. 1, 174–180. MR 1367618, DOI 10.1006/jcta.1996.0013
- John D. Wiltshire-Gordon, Uniformly presented vector spaces, arXiv:1406.0786, 2014.
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: 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