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Register a new userOn decomposition numbers with Jantzen filtration of cyclotomic $q$-Schur algebras
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Authors
Kentaro Wada
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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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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 $H_{n_k,r_k}$.
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.
The cyclotomic Birman-Murakami-Wenzl (BMW) algebras B_n^k, introduced by R. Haring-Oldenburg, are a generalisation of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (aka Ariki-Koike algebras) and type B knot theory. In this paper, we prove the algebra is free and of rank k^n (2n-1)!! over ground rings with parameters satisfying so-called admissibility conditions. These conditions are necessary in order for these results to hold and originally arise from the representation theory of B_2^k, which is analysed by the authors in a previous paper. Furthermore, we obtain a geometric realisation of B_n^k as a cyclotomic version of the Kauffman tangle algebra, in terms of affine n-tangles in the solid torus, and produce explicit bases that may be described both algebraically and diagrammatically. The admissibility conditions are the most general offered in the literature for which these results hold; they are necessary and sufficient for all results for general n.
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