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Register a new userNew realization of cyclotomic $q$-Schur algebras I
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Authors
Kentaro Wada
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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.
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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.
For a complex finite-dimensional simple Lie algebra $mathfrak{g}$, we introduce the notion of Q-datum, which generalizes the notion of a Dynkin quiver with a height function from the viewpoint of Weyl group combinatorics. Using this notion, we develop a unified theory describing the twisted Auslander-Reiten quivers and the twisted adapted classes introduced in [O.-Suh, J. Algebra, 2019] with an appropriate notion of the generalized Coxeter elements. As a consequence, we obtain a combinatorial formula expressing the inverse of the quantum Cartan matrix of $mathfrak{g}$, which generalizes the result of [Hernandez-Leclerc, J. Reine Angew. Math., 2015] in the simply-laced case. We also find several applications of our combinatorial theory of Q-data to the finite-dimensional representation theory of the untwisted quantum affine algebra of $mathfrak{g}$. In particular, in terms of Q-data and the inverse of the quantum Cartan matrix, (i) we give an alternative description of the block decomposition results due to [Chari-Moura, Int. Math. Res. Not., 2005] and [Kashiwara-Kim-O.-Park, arXiv:2003.03265], (ii) we present a unified (partially conjectural) formula of the denominators of the normalized R-matrices between all the Kirillov-Reshetikhin modules, and (iii) we compute the invariants $Lambda(V,W)$ and $Lambda^infty(V, W)$ introduced in [Kashiwara-Kim-O.-Park, Compos. Math., 2020] for each pair of simple modules $V$ and $W$.
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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