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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Hecke algebras, $U_{q}sl_{n}$, and
the Donald-Flanigan conjecture for $S_{n}$


Authors: Murray Gerstenhaber and Mary E. Schaps
Journal: Trans. Amer. Math. Soc. 349 (1997), 3353-3371
MSC (1991): Primary 20C30; Secondary 17B35, 17B37, 20F99
MathSciNet review: 1390035
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Abstract: The Donald-Flanigan conjecture asserts that the integral group ring $\mathbb {Z}G$ of a finite group $G$ can be deformed to an algebra $A$ over the power series ring $\mathbb {Z}[[t]]$ with underlying module $\mathbb {Z}G[[t]]$ such that if $p$ is any prime dividing $\#G$ then $A\otimes _{\mathbb {Z}[[t]]}\overline {\mathbb {F}_{p}((t))}$ is a direct sum of total matric algebras whose blocks are in natural bijection with and of the same dimensions as those of $\mathbb {C}G.$ We prove this for $G = S_{n}$ using the natural representation of its Hecke algebra $\mathcal {H}$ by quantum Yang-Baxter matrices to show that over $\mathbb {Z}[q]$ localized at the multiplicatively closed set generated by $q$ and all $i_{q^{2}} = 1+q^{2} + q^{4} + \dots + q^{2(i-1)}, i = 1,2,\dots , n$, the Hecke algebra becomes a direct sum of total matric algebras. The corresponding ``canonical" primitive idempotents are distinct from Wenzl's and in the classical case ($q=1$), from those of Young.


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Additional Information

Murray Gerstenhaber
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Email: mgersten@mail.sas.upenn.edu or murray@math.upenn.edu

Mary E. Schaps
Affiliation: Department of Mathematics and Computer Science, Bar Ilan University, Ramat-Gan 52900, Israel
Email: mschaps@macs.biu.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01761-3
PII: S 0002-9947(97)01761-3
Keywords: Hecke algebra, representations, symmetric group, deformations, quantization, Donald-Flanigan conjecture
Received by editor(s): February 1, 1994
Received by editor(s) in revised form: February 26, 1996
Additional Notes: Research of the first author was partially supported by a grant from the NSA
Article copyright: © Copyright 1997 American Mathematical Society