A modified Brauer algebra as centralizer algebra of the unitary group
HTML articles powered by AMS MathViewer
- by Alberto Elduque PDF
- Trans. Amer. Math. Soc. 356 (2004), 3963-3983 Request permission
Abstract:
The centralizer algebra of the action of $U(n)$ on the real tensor powers $\otimes _\mathbb {R}^r V$ of its natural module, $V=\mathbb {C}^n$, is described by means of a modification in the multiplication of the signed Brauer algebras. The relationships of this algebra with the invariants for $U(n)$ and with the decomposition of $\otimes _\mathbb {R}^r V$ into irreducible submodules is considered.References
- Elsa Abbena and Sergio Garbiero, Almost Hermitian homogeneous structures, Proc. Edinburgh Math. Soc. (2) 31 (1988), no. 3, 375–395. MR 969067, DOI 10.1017/S0013091500006775
- Georgia Benkart, Manish Chakrabarti, Thomas Halverson, Robert Leduc, Chanyoung Lee, and Jeffrey Stroomer, Tensor product representations of general linear groups and their connections with Brauer algebras, J. Algebra 166 (1994), no. 3, 529–567. MR 1280591, DOI 10.1006/jabr.1994.1166
- Joan S. Birman and Hans Wenzl, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. 313 (1989), no. 1, 249–273. MR 992598, DOI 10.1090/S0002-9947-1989-0992598-X
- R. Brauer, On algebras which are connected with semisimple Lie groups, Ann. of Math. 38 (1937), 857–872.
- P. Fortuny, and P. Martínez Gadea, On the classification theorems of almost-Hermitian or homogeneous Kähler structures, to appear in Rocky Mountain. J. Math.
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
- Alfred Gray and Luis M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. (4) 123 (1980), 35–58. MR 581924, DOI 10.1007/BF01796539
- Tom Halverson and Arun Ram, Characters of algebras containing a Jones basic construction: the Temperley-Lieb, Okada, Brauer, and Birman-Wenzl algebras, Adv. Math. 116 (1995), no. 2, 263–321. MR 1363766, DOI 10.1006/aima.1995.1068
- Phil Hanlon and David Wales, On the decomposition of Brauer’s centralizer algebras, J. Algebra 121 (1989), no. 2, 409–445. MR 992775, DOI 10.1016/0021-8693(89)90076-8
- Nagayoshi Iwahori, Some remarks on tensor invariants of $\textrm {O}(n), \textrm {U}(n),\textrm {Sp}(n)$, J. Math. Soc. Japan 10 (1958), 145–160. MR 124410, DOI 10.2969/jmsj/01020145
- Michio Jimbo, A $q$-analogue of $U({\mathfrak {g}}{\mathfrak {l}}(N+1))$, Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), no. 3, 247–252. MR 841713, DOI 10.1007/BF00400222
- Robert Leduc and Arun Ram, A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras: the Brauer, Birman-Wenzl, and type A Iwahori-Hecke algebras, Adv. Math. 125 (1997), no. 1, 1–94. MR 1427801, DOI 10.1006/aima.1997.1602
- Jun Murakami, The representations of the $q$-analogue of Brauer’s centralizer algebras and the Kauffman polynomial of links, Publ. Res. Inst. Math. Sci. 26 (1990), no. 6, 935–945. MR 1079902, DOI 10.2977/prims/1195170569
- M. Parvathi and M. Kamaraj, Signed Brauer’s algebras, Comm. Algebra 26 (1998), no. 3, 839–855. MR 1606174, DOI 10.1080/00927879808826168
- M. Parvathi and C. Selvaraj, Signed Brauer’s algebras as centralizer algebras, Comm. Algebra 27 (1999), no. 12, 5985–5998. MR 1726289, DOI 10.1080/00927879908826803
- Issai Schur, Gesammelte Abhandlungen. Band I, Springer-Verlag, Berlin-New York, 1973 (German). Herausgegeben von Alfred Brauer und Hans Rohrbach. MR 0462891, DOI 10.1007/978-3-642-61947-2
- Issai Schur, Gesammelte Abhandlungen. Band I, Springer-Verlag, Berlin-New York, 1973 (German). Herausgegeben von Alfred Brauer und Hans Rohrbach. MR 0462891, DOI 10.1007/978-3-642-61947-2
- John R. Stembridge, Rational tableaux and the tensor algebra of $\textrm {gl}_n$, J. Combin. Theory Ser. A 46 (1987), no. 1, 79–120. MR 899903, DOI 10.1016/0097-3165(87)90077-X
- Hans Wenzl, On the structure of Brauer’s centralizer algebras, Ann. of Math. (2) 128 (1988), no. 1, 173–193. MR 951511, DOI 10.2307/1971466
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
Additional Information
- Alberto Elduque
- Affiliation: Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain
- MR Author ID: 208418
- Email: elduque@unizar.es
- Received by editor(s): June 9, 2003
- Published electronically: May 10, 2004
- Additional Notes: This research was supported by the Spanish Ministerio de Ciencia y Tecnología and FEDER (BFM 2001-3239-C03-03)
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 3963-3983
- MSC (2000): Primary 20G05, 17B10
- DOI: https://doi.org/10.1090/S0002-9947-04-03602-5
- MathSciNet review: 2058514