Uno's conjecture on representation types of Hecke algebras

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Let KK be a field, let qKq\in K be an invertible element with q1q\ne 1, and let (W,S)(W,S) be a finite Weyl group. Let Hq(W){\mathcal H}_q(W) be the one-parameter Hecke algebra associated to WW, with quadratic relations (Ts+1)(Tsq)=0(T_s+1)(T_s-q)=0 for sSs\in S, and let PW(x)=wWxl(w)P_W(x)=\sum_{w\in W}x^{l(w)} be its Poincaré polynomial. Assume that KK is a splitting field of Hq(W){\mathcal H}_q(W). Uno's conjecture. The algebra Hq(W){\mathcal H}_q(W) is of finite representation type and not semisimple if and only if qq is a simple root of PW(x)=0P_W(x)=0, equivalently, if and only if (xq)2(x-q)^2 does not divide PW(x)P_W(x). The conjecture gives a qq-analogue of the criterion for a group algebra of a finite Weyl group to have finite representation type. The source paper states that it proves Uno's conjecture for all Hecke algebras of classical type; its status is otherwise not resolved by the supplied text.

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Sources & referencesView supporting material

Primary source

Susumu Ariki, “Uno's conjecture on representation types of Hecke algebras”, arXiv:math/0108176 (2003).

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