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

Gerstenhaber, Murray; Schaps, Mary

doi:10.1090/S0002-9947-97-01761-3

Hecke algebras, $U_qsl_n$, and the Donald-Flanigan conjecture for $S_n$
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by Murray Gerstenhaber and Mary E. Schaps PDF

Trans. Amer. Math. Soc. 349 (1997), 3353-3371 Request permission

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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