by Jakobsen, Hans Plesner; Zhang, Hechun

In the present article, a basis of the coordinate algebra of the multi-parameter quantized matrix is constructed by using an elementary method due to Lusztig. The construction depends heavily on an anti-automorphism, the bar action. The exponential nature of the bar action is derived which provides an inductive way to compute the basis elements. By embedding the basis into the dual basis of Lusztig's canonical basis of $U_q(n^-)$, the positivity properties of the basis as well as the positivity properties of the canonical basis of the modified quantum enveloping algebra of type $A$, which has been conjectured by Lusztig, are proved.

DOI: 10.1007/s10468-006-9017-2
Online Date: 6/29/2006
Print publication date: 6/1/2006
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by Kawata, Shigeto

Let $G$ be a finite group, ${\mathcal O}$ a complete discrete valuation ring of characteristic zero with residue class field ${\mathcal O} / \pi {\mathcal O}$ of characteristic $p > 0$, and $B$ a block of the group ring ${\mathcal O}G$. Suppose that $B$ is of infinite representation type and ${\mathcal O}$ is sufficiently large to satisfy certain conditions. Let $\Gamma({\mathcal O}G)$ be the Auslander–Reiten quiver of ${\mathcal O}G$ and $\Theta$ a connected component of $\Gamma({\mathcal O}G)$. In this paper, we show that if $\Theta$ contains some Heller lattices then the tree class of the stable part of $\Theta$ is $A_{\infty}$. Also, we show that $B$ has infinitely many components of type ${\bf{Z}}A_{\kern1.15pt\infty}$ if a defect group of $B$ is neither cyclic nor a Klein four group.

DOI: 10.1007/s10468-006-9025-2
Online Date: 6/28/2006
Print publication date: 10/1/2006
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by Holmes, Randall R.; Zhang, Chaowen

The simple modules with homogeneous characters are considered, their dimension formulas are determined.

DOI: 10.1007/s10468-006-9023-4
Online Date: 6/28/2006
Print publication date: 6/1/2006
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by Mitsuhashi, Hideo

In this paper, we establish Schur–Weyl reciprocity between the quantum general super Lie algebra $U_q^\sigma\big{(}{\mathfrak{g}\mathfrak{l}}(m,n)\big{)}$ and the Iwahori–Hecke algebra $\mathcal{H}_{\mathbb{Q}(q),r}(q)$. We introduce the sign $q$-permutation representation of $\mathcal{H}_{\mathbb{Q}(q),r}(q)$ on the tensor space $V^{{\otimes}r}$ of $(m+n)$ dimensional $\mathbb{Z}_{\,2}$-graded $\mathbb{Q}(q)$-vector space $V=V_{\bar{0}}{\oplus}V_{\bar{1}}$. This action commutes with that of $U_q^\sigma\big{(}{\mathfrak{g}\mathfrak{l}}(m,n)\big{)}$ derived from the vector representation on $V$. Those two subalgebras of $\operatorname{End}_{\mathbb{Q}(q)}(V^{{\otimes}r})$ satisfy Schur–Weyl reciprocity. As special cases, we obtain the super case ($q{\rightarrow}1$), and the quantum case ($n=0$). Hence this result includes both the super case and the quantum case, and unifies those two important cases.

DOI: 10.1007/s10468-006-9014-5
Online Date: 6/20/2006
Print publication date: 6/1/2006
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