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We develop algebraic and geometrical approaches toward canonical bases for affine q-Schur algebras of arbitrary type introduced in this paper. A duality between an affine q-Schur algebra and a corresponding affine Hecke algebra is established. We introduce an inner product on the affine q-Schur algebra, with respect to which the canonical basis is shown to be positive and almost orthonormal. We then formulate the cells and asymptotic forms for affine q-Schur algebras, and develop their basic properties analogous to the cells and asymptotic forms for affine Hecke algebras established by Lusztig. The results on cells and asymptotic algebras are also valid for q-Schur algebras of arbitrary finite type.
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We introduce a Lie algebra $mathfrak{g}_{mathbf{Q}}(mathbf{m})$ and an associative algebra $mathcal{U}_{q,mathbf{Q}}(mathbf{m})$ associated with the Cartan data of $mathfrak{gl}_m$ which is separated into $r$ parts with respect to $mathbf{m}=(m_1, dots, m_r)$ such that $m_1+ dots + m_r =m$. We show that the Lie algebra $mathfrak{g}_{mathbf{Q}} (mathbf{m})$ is a filtered deformation of the current Lie algebra of $mathfrak{gl}_m$, and we can regard the algebra $mathcal{U}_{q, mathbf{Q}}(mathbf{m})$ as a $q$-analogue of $U(mathfrak{g}_{mathbf{Q}}(mathbf{m}))$. Then, we realize a cyclotomic $q$-Schur algebra as a quotient algebra of $mathcal{U}_{q, mathbf{Q}}(mathbf{m})$ under a certain mild condition. We also study the representation theory for $mathfrak{g}_{mathbf{Q}}(mathbf{m})$ and $mathcal{U}_{q,mathbf{Q}}(mathbf{m})$, and we apply them to the representations of the cyclotomic $q$-Schur algebras.
In this paper the authors investigate the $q$-Schur algebras of type B that were constructed earlier using coideal subalgebras for the quantum group of type A. The authors present a coordinate algebra type construction that allows us to realize these $q$-Schur algebras as the duals of the $d$th graded components of certain graded coalgebras. Under suitable conditions an isomorphism theorem is proved that demonstrates that the representation theory reduces to the $q$-Schur algebra of type A. This enables the authors to address the questions of cellularity, quasi-hereditariness and representation type of these algebras. Later it is shown that these algebras realize the $1$-faithful quasi hereditary covers of the Hecke algebras of type B. As a further consequence, the authors demonstrate that these algebras are Morita equivalent to Rouquiers finite-dimensional algebras that arise from the category ${mathcal O}$ for rational Cherednik algebras for the Weyl group of type B. In particular, we have introduced a Schur-type functor that identifies the type B Knizhnik-Zamolodchikov functor.
We give a necessary and sufficient condition on parameters for Ariki-Koike algebras (resp. cyclotomic q-Schur algebras) to be of finite representation type.
Let $S$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $H_{n,r}$ of rank $n$, introduced by Dipper-James-Mathas. For each $p = (r_1, ..., r_g)$ such that $r_1 + ... + r_g = r$, we define a subalgebra $S^p$ of $S$ and its quotient algebra $bar S^p$. It is shown that $S^p$ is a standardly based algebra and $bar S^p$ is a cellular algebra. By making use of these algebras, we show that certain decomposition numbers for $S$ can be expressed as a product of decomposition numbers for cyclotomic $q$-Schur algebras associated to smaller Ariki_koike algebras $H_{n_k,r_k}$.
Let $Sc(vL)$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $He_{n,r}$, introduced by Dipper-James-Mathas. In this paper, we consider $v$-decomposition numbers of $Sc(vL)$, namely decomposition numbers with respect to the Jantzen filtrations of Weyl modules. We prove, as a $v$-analogue of the result obtained by Shoji-Wada, a product formula for $v$-decomposition numbers of $Sc(vL)$, which asserts that certain $v$-decomposition numbers are expressed as a product of $v$-decomposition numbers for various cyclotomic $q$-Schur algebras associated to Ariki-koike algebras $He_{n_i,r_i}$ of smaller rank. Moreover we prove a similar formula for $v$-decomposition numbers of $He_{n,r}$ by using a Schur functor.
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