\(G(\ell ,k,d)\) -modules via groupoids

详细信息    查看全文

• 作者：Volodymyr Mazorchuk ; Catharina Stroppel
• 关键词：Schur–Weyl duality ; Wreath product ; Simple module ; Groupoid
• 刊名：Journal of Algebraic Combinatorics
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：43
• 期：1
• 页码：11-32
• 全文大小：708 KB
• 参考文献：1.Adin, R., Postnikov, A., Roichman, Y.: Combinatorial Gelfand models. J. Algebra 320(3), 1311–1325 (2008)MATH MathSciNet CrossRef
  2.Adin, R., Postnikov, A., Roichman, Y.: A Gelfand model for wreath products. Isr. J. Math. 179, 381–402 (2010)MATH MathSciNet CrossRef
  3.Ariki, S., Terasoma, T., Yamada, H.: Schur–Weyl reciprocity for the Hecke algebra of \(({\mathbb{Z}}/r{\mathbb{Z}})\wr S_n\) . J. Algebra 178(2), 374–390 (1995)MATH MathSciNet CrossRef
  4.Bao, H., Wang, W.: A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs. arXiv:​1310.​0103
  5.Can, H.: Representations of the generalized symmetric groups. Beiträge Algebra Geom. 37(2), 289–307 (1996)MATH MathSciNet
  6.Caselli, F., Fulci, R.: Refined Gelfand models for wreath products. Eur. J. Comb. 32(2), 198–216 (2011)MATH MathSciNet CrossRef
  7.Caselli, F., Marberg, E.: Isomorphisms, automorphisms, and generalized involution models of projective reflection groups. Isr. J. Math. 199(1), 433–483 (2014)MATH MathSciNet CrossRef
  8.Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F.: Representation theory of wreath products of finite groups. J. Math. Sci. 156(1), 44–55 (2009)MATH MathSciNet CrossRef
  9.Ehrig, M., Stroppel, C.: Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality. Preprint arXiv:​1310.​1972
  10.Ehrig, M., Stroppel, C.: 2-row Springer fibres and Khovanov diagram algebras for type D. Can. J. Math. Preprint arXiv:​1209.​49989 (to appear)
  11.Ganyushkin, O., Mazorchuk, V.: Classical finite transformation semigroups. An introduction, Algebra and Applications, 9. Springer, (2009)
  12.Goodman, R., Wallach, N.: Symmetry, Representations, and Invariants. Graduate Texts in Mathematics, vol. 255. Springer, Berlin (2009)
  13.Grood, C.: A Specht module analog for the rook monoid. Electron. J. Combin. 9(1) Research Paper 2 (2002)
  14.Halverson, T., Ram, A.: \(q\) -Rook monoid algebras, Hecke algebras, and Schur–Weyl duality. J. Math. Sci. 121(3), 2419–2436 (2004)MathSciNet CrossRef
  15.Halverson, T., Reeks, M.: Gelfand models for diagram algebras. J. Algebraic Comb. 41(2), 229–255 (2015)MATH MathSciNet CrossRef
  16.Hu, J.: Schur–Weyl reciprocity between quantum groups and Hecke algebras of type \(G(r,1, n)\) . Math. Z. 238(3), 505–521 (2001)MATH MathSciNet CrossRef
  17.Hu, J., Shoji, T.: Schur–Weyl reciprocity between quantum groups and Hecke algebras of type \(G(p, p, n)\) . J. Algebra 298(1), 215–237 (2006)MATH MathSciNet CrossRef
  18.Inglis, N., Richardson, R., Saxl, J.: An explicit model for the complex representations of \(S_n\) . Arch. Math. (Basel) 54(3), 258–259 (1990)MATH MathSciNet CrossRef
  19.James, G., Kerber, A.: The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and Its Applications, vol. 16. Addison-Wesley, Reading (1981)
  20.Kerber, A.: Zur Darstellungstheorie von Kranzprodukten. Can. J. Math. 20, 665–672 (1968)MATH MathSciNet CrossRef
  21.Kraft, H., Procesi, C.: Classical invariant theory, a primer. Manuscript (1996). Available at: http://​jones.​math.​unibas.​ch/​~kraft/​Papers/​KP-Primer
  22.Kudryavtseva, G., Mazorchuk, V.: Combinatorial Gelfand models for some semigroups and \(q\) -rook monoid algebras. Proc. Edinb. Math. Soc. (2) 52(3), 707–718 (2009)MATH MathSciNet CrossRef
  23.Kudryavtseva, G., Mazorchuk, V.: On presentations of Brauer-type monoids. Cent. Eur. J. Math. 4(3), 413–434 (2006)MATH MathSciNet CrossRef
  24.Marin, I.: Branching properties for the groups \(G(de, e, r)\) . J. Algebra 323(4), 966–982 (2010)MATH MathSciNet CrossRef
  25.Mazorchuk, V.: Combinatorial Gelfand models for semisimple diagram algebras. Milan J. Math. 81(2), 385–396 (2013)MATH MathSciNet CrossRef
  26.Moon, D.: A diagram realization of complex reflection groups and Schur–Weyl dualities. Trends in mathematics. Inf. Center Math. Sci. 8(2), 119–127 (2005)
  27.Osima, M.: On the representations of the generalized symmetric group. Math. J. Okayama Univ. 4, 39–56 (1954)MATH MathSciNet
  28.Regev, A.: Double centralizing theorems for wreath product. Contemp. Math. 34, 67–72 (1984)MathSciNet CrossRef
  29.Saeed-Ul-Islam, M.: Irreducible representations of the generalized symmetric group \(B_n^m\) . Glasgow Math. J. 29(1), 1–6 (1987)MATH MathSciNet CrossRef
  30.Sagan, B.: The Symmetric Group. Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd edn. Graduate Texts in Mathematics, vol. 203. Springer, Berlin (2001)
  31.Sakamoto, M., Shoji, T.: Schur–Weyl reciprocity for Ariki–Koike algebras. J. Algebra 221(1), 293–314 (1999)MATH MathSciNet CrossRef
  32.Sartori, A., Stroppel, C.: Coidal subalgebras: PBW theorem and representation theory I. Preprint
  33.Schur, I.: Über die rationalen Darstellungen der allgemeinen linearen Gruppe, pp. 58–75. Sitzungsberichte Akad, Berlin (1927). Reprinted in: I. Schur. Gesammelte Abhandlungen III, pp. 68–85. Springer, Berlin (1973)
  34.Schur, I.: Über eine Klasse von Matrizen die sich einer gegebenen Matrix zuordnen lassen. Thesis, Berlin (1901). Reprinted in: I. Schur, Gesammelte Abhandlungen I, pp. 1–70. Springer, Berlin (1973)
  35.Serre, J.-P.: Linear Representations of Finite Groups. Graduate Texts in Mathematics, vol. 42. Springer, Berlin (1977)
  36.Shoji, T.: A Frobenius formula for the characters of Ariki–Koike algebras. J. Algebra 226(2), 818–856 (2000)MATH MathSciNet CrossRef
  37.Solomon, L.: Representations of the rook monoid. J. Algebra 256(2), 309–342 (2002)MATH MathSciNet CrossRef
  38.Specht, W.: Eine Verallgemeinerung der symmetrischen Gruppe. Berl. Semin. 1, 1–32 (1932)
  39.Weyl, H.: The Classical Groups: Their Invariants and Representations. Princeton University Press, Princeton (1939)
  
• 作者单位：Volodymyr Mazorchuk (1)
  Catharina Stroppel (2)
  
  1. Department of Mathematics, Uppsala University, Box 480, 751 06, Uppsala, Sweden
  2. Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Convex and Discrete Geometry
  Order, Lattices and Ordered Algebraic Structures
  Computer Science, general
  Group Theory and Generalizations
  
• 出版者：Springer U.S.
• ISSN：1572-9192

文摘

In this note, we describe a seemingly new approach to the complex representation theory of the wreath product \(G\wr S_d\), where G is a finite abelian group. The approach is motivated by an appropriate version of Schur–Weyl duality. We construct a combinatorially defined groupoid in which all endomorphism algebras are direct products of symmetric groups and prove that the groupoid algebra is isomorphic to the group algebra of \(G\wr S_d\). This directly implies a classification of simple modules. As an application, we get a Gelfand model for \(G\wr S_d\) from the classical involutive Gelfand model for the symmetric group. We describe the Schur–Weyl duality which motivates our approach and relate it to various Schur–Weyl dualities in the literature. Finally, we discuss an extension of these methods to all complex reflection groups of type \(G(\ell ,k,d)\).