Let $W$ be a finite Weyl group of classical type which may not beirreducible, $F$ an algebraically closed field, $q$ an invertible element of$F$. We denote by $\mathcal H_W(q)$ the associated Hecke algebra. If $q=1$ thenit is $FW$ and we know the representation type. Thus, we assume that $q\ne 1$.Let $P_W(x)$ be the Poincare polynomial of $W$. It is well-known that $\mathcalH_W(q)$ is semisimple if and only if $x-q$ does not divide $P_W(x)$. We showthat the similar results hold for finiteness, tameness and wildness. In otherwords, the Poincare polynomial governs the representation type of $\mathcalH_W(q)$ completely. Note that the finiteness result was already given in theauthor's previous papers, some of which were written with Andrew Mathas. The proof uses the Fock space theory, which was developed for proving the LLTconjecture (see AMS Univ. Lec. Ser. 26), the Specht module theory, which wasdeveloped by Dipper, James and Murphy in this case, and results from the theoryof finite dimensional algebras.
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机译：假设$ W $是一个经典的有限Weyl群，它可能是不可约的，$ F $是一个代数封闭域，$ q $是$ F $的可逆元素。我们用\\数学H_W（q）$表示相关的Hecke代数。如果$ q = 1 $ thenit是$ FW $，我们知道表示类型。因此，我们假设$ q \ ne 1 $。让$ P_W（x）$是$ W $的Poincare多项式。众所周知，当且仅当$ x-q $不除以$ P_W（x）$时，$ \ mathcalH_W（q）$是半简单的。我们表明，类似的结果适用于有限性，驯服性和野性。换句话说，庞加莱多项式完全控制$ \ mathcalH_W（q）$的表示类型。注意，有限性结果已经在作者以前的论文中给出，其中一些论文是由安德鲁·马塔斯（Andrew Mathas）撰写的。该证明使用了为证明LLT猜想而开发的Fock空间理论（请参见AMS Univ。Lec。Ser。26），在这种情况下由Dipper，James和Murphy开发的Specht模块理论，它是有限理论的结果维代数。
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