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Register a new userSchur algebras and quantum symmetric pairs with unequal parameters
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We study the (quantum) Schur algebras of type B/C corresponding to the Hecke algebras with unequal parameters. We prove that the Schur algebras afford a stabilization construction in the sense of Beilinson-Lusztig-MacPherson that constructs a multiparameter upgrade of the quantum symmetric pair coideal subalgebras of type A III/IV with no black nodes. We further obtain the canonical basis of the Schur/coideal subalgebras, at the specialization associated to any weight function. These bases are the counterparts of Lusztigs bar-invariant basis for Hecke algebras with unequal parameters. In the appendix we provide an algebraic version of a type D Beilinson-Lusztig-MacPherson construction which is first introduced by Fan-Li from a geometric viewpoint.
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We establish a three-parameter Schur duality between the affine Hecke algebra of type C and a coideal subalgebra of quantum affine $mathfrak{sl}_n$. At the equal parameter specializations, we obtain Schur dualities of types BCD.
Let $U_q(mathfrak{g})$ be a twisted affine quantum group of type $A_{N}^{(2)}$ or $D_{N}^{(2)}$ and let $mathfrak{g}_{0}$ be the finite-dimensional simple Lie algebra of type $A_{N}$ or $D_{N}$. For a Dynkin quiver of type $mathfrak{g}_{0}$, we define a full subcategory ${mathcal C}_{Q}^{(2)}$ of the category of finite-dimensional integrable $U_q(mathfrak{g})$-modules, a twisted version of the category ${mathcal C}_{Q}$ introduced by Hernandez and Leclerc. Applying the general scheme of affine Schur-Weyl duality, we construct an exact faithful KLR-type duality functor ${mathcal F}_{Q}^{(2)}: Rep(R) rightarrow {mathcal C}_{Q}^{(2)}$, where $Rep(R)$ is the category of finite-dimensional modules over the quiver Hecke algebra $R$ of type $mathfrak{g}_{0}$ with nilpotent actions of the generators $x_k$. We show that ${mathcal F}_{Q}^{(2)}$ sends any simple object to a simple object and induces a ring isomorphism $K(Rep(R)) simeq K({mathcal C}_{Q}^{(2)})$.
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.
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.
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 $H_{n_k,r_k}$.
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