The partition algebra and the Kronecker coefficients
HTML articles powered by AMS MathViewer
- by C. Bowman, M. De Visscher and R. Orellana PDF
- Trans. Amer. Math. Soc. 367 (2015), 3647-3667 Request permission
Abstract:
We propose a new approach to study the Kronecker coefficients by using the Schur–Weyl duality between the symmetric group and the partition algebra. We explain the limiting behaviour and associated bounds in the context of the partition algebra. Our analysis leads to a uniform description of the reduced Kronecker coefficients when one of the indexing partitions is a hook or a two-part partition.References
- D. J. Benson, Representations and cohomology. I, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1998. Basic representation theory of finite groups and associative algebras. MR 1644252
- C. Bessenrodt and A. Kleshchev, On Kronecker products of complex representations of the symmetric and alternating groups, Pacific J. Math. 190 (1999), no. 2, 201–223. MR 1722888, DOI 10.2140/pjm.1999.190.201
- J. Blasiak, Kronecker coefficients of one hook shape, arxiv:1209.2209v2, 2012.
- C. M. Ballantine and R. C. Orellana, On the Kronecker product $s_{(n-p,p)}\ast s_\lambda$, Electron. J. Combin. 12 (2005), Research Paper 28, 26. MR 2156682
- Emmanuel Briand, Rosa Orellana, and Mercedes Rosas, The stability of the Kronecker product of Schur functions, J. Algebra 331 (2011), 11–27. MR 2774644, DOI 10.1016/j.jalgebra.2010.12.026
- 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
- Vlastimil Dlab and Claus Michael Ringel, The module theoretical approach to quasi-hereditary algebras, Representations of algebras and related topics (Kyoto, 1990) London Math. Soc. Lecture Note Ser., vol. 168, Cambridge Univ. Press, Cambridge, 1992, pp. 200–224. MR 1211481
- Yoav Dvir, On the Kronecker product of $S_n$ characters, J. Algebra 154 (1993), no. 1, 125–140. MR 1201916, DOI 10.1006/jabr.1993.1008
- A. M. Garsia and J. Remmel, Shuffles of permutations and the Kronecker product, Graphs Combin. 1 (1985), no. 3, 217–263. MR 951014, DOI 10.1007/BF02582950
- Roe Goodman and Nolan R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998. MR 1606831
- Tom Halverson and Arun Ram, Partition algebras, European J. Combin. 26 (2005), no. 6, 869–921. MR 2143201, DOI 10.1016/j.ejc.2004.06.005
- Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
- V. F. R. Jones, The Potts model and the symmetric group, Subfactors (Kyuzeso, 1993) World Sci. Publ., River Edge, NJ, 1994, pp. 259–267. MR 1317365
- A. Klyachko, Quantum marginal problem and representations of the symmetric group, arXiv:quant-ph/0409113 (2004).
- A. Lascoux, Produit de Kronecker des représentations du groupe symétrique, Séminaire d’Algèbre Paul Dubreil et Marie-Paule Malliavin, 32ème année (Paris, 1979) Lecture Notes in Math., vol. 795, Springer, Berlin, 1980, pp. 319–329 (French). MR 582085
- 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
- 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
- Paul Martin, Potts models and related problems in statistical mechanics, Series on Advances in Statistical Mechanics, vol. 5, World Scientific Publishing Co., Inc., Teaneck, NJ, 1991. MR 1103994, DOI 10.1142/0983
- Paul Martin, The structure of the partition algebras, J. Algebra 183 (1996), no. 2, 319–358. MR 1399030, DOI 10.1006/jabr.1996.0223
- Paul Martin and Hubert Saleur, On an algebraic approach to higher-dimensional statistical mechanics, Comm. Math. Phys. 158 (1993), no. 1, 155–190. MR 1243720, DOI 10.1007/BF02097236
- 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
- Francis D. Murnaghan, On the analysis of the Kronecker product of irreducible representations of $S_n$, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 515–518. MR 70640, DOI 10.1073/pnas.41.7.515
- Jean-Yves Thibon, Hopf algebras of symmetric functions and tensor products of symmetric group representations, Internat. J. Algebra Comput. 1 (1991), no. 2, 207–221. MR 1128013, DOI 10.1142/S0218196791000134
- Ernesto Vallejo, Stability of Kronecker products of irreducible characters of the symmetric group, Electron. J. Combin. 6 (1999), Research Paper 39, 7. MR 1725703
Additional Information
- C. Bowman
- Affiliation: Institut de Mathématiques de Jussieu, 5 rue du Thomas Mann, 75013, Paris, France
- MR Author ID: 922280
- Email: Bowman@math.jussieu.fr
- M. De Visscher
- Affiliation: Centre for Mathematical Science, City University London, Northampton Square, London, EC1V 0HB, England
- MR Author ID: 703480
- Email: Maud.Devisscher.1@city.ac.uk
- R. Orellana
- Affiliation: Department of Mathematics, 6188 Kemeny Hall, Dartmouth College, Hanover, New Hampshire 03755
- Email: Rosa.C.Orellana@dartmouth.edu
- Received by editor(s): March 6, 2013
- Received by editor(s) in revised form: June 10, 2013, and July 4, 2013
- Published electronically: December 11, 2014
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 3647-3667
- MSC (2010): Primary 20C30, 05E10
- DOI: https://doi.org/10.1090/S0002-9947-2014-06245-4
- MathSciNet review: 3314819