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Authors
Karel Casteels
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We study, from a combinatorial viewpoint, the quantized coordinate ring of mxn matrices over an infinite field K (also called quantum matrices) and its torus-invariant prime ideals. The first part of this paper shows that this algebra, traditionally defined by generators and relations, can be seen as subalgebra of a quantum torus by using paths in a certain directed graph. Roughly speaking, we view each generator of quantum matrices as a sum over paths in the graph, each path being assigned an element of the quantum torus. The quantum matrices relations then arise naturally by considering intersecting paths. This viewpoint is closely related to Cauchons deleting-derivations algorithm. The second part of this paper is to apply the paths viewpoint to the theory of torus-invariant prime ideals of quantum matrices. We prove a conjecture of Goodearl and Lenagan that all such prime ideals, when the quantum parameter q is a non-root of unity, have generating sets consisting of quantum minors. Previously, this result was known to hold only for char(K)=0 and q transcendental over Q. Our strategy is to show that the quantum minors in a given torus-invariant ideal form a Grobner basis.
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We present a combinatorial method to determine the dimension of $C{H}$-strata in the algebra of $mtimes n$ quantum matrices $Oq$ as follows. To a given $C{H}$-stratum we associate a certain permutation via the notion of pipe-dreams. We show that the dimension of the $C{H}$-stratum is precisely the number of odd cycles in this permutation. Using this result, we are able to give closed formulas for the trivariate generating function that counts the $d$-dimensional $C{H}$-strata in $Oq$. Finally, we extract the coefficients of this generating function in order to settle conjectures proposed by the first and third named authors cite{bldim,bll} regarding the asymptotic proportion of $d$-dimensional $C{H}$-strata in $Oq$.
We take a graph theoretic approach to the problem of finding generators for those prime ideals of $mathcal{O}_q(mathcal{M}_{m,n}(mathbb{K}))$ which are invariant under the torus action ($mathbb{K}^*)^{m+n}$. Launois cite{launois3} has shown that the generators consist of certain quantum minors of the matrix of canonical generators of $mathcal{O}_q(mathcal{M}_{m,n}(mathbb{K}))$ and in cite{launois2} gives an algorithm to find them. In this paper we modify a classic result of Lindstr{o}m cite{lind} and Gessel-Viennot~cite{gv} to show that a quantum minor is in the generating set for a particular ideal if and only if we can find a particular set of vertex-disjoint directed paths in an associated directed graph.
Let $lambda$ be a (level-zero) dominant integral weight for an untwisted affine Lie algebra, and let $mathrm{QLS}(lambda)$ denote the quantum Lakshmibai-Seshadri (QLS) paths of shape $lambda$. For an element $w$ of a finite Weyl group $W$, the specializations at $t = 0$ and $t = infty$ of the nonsymmetric Macdonald polynomial $E_{w lambda}(q, t)$ are explicitly described in terms of QLS paths of shape $lambda$ and the degree function defined on them. Also, for (level-zero) dominant integral weights $lambda$, $mu$, we have an isomorphism $Theta : mathrm{QLS}(lambda + mu) rightarrow mathrm{QLS}(lambda) otimes mathrm{QLS}(mu)$ of crystals. In this paper, we study the behavior of the degree function under the isomorphism $Theta$ of crystals through the relationship between semi-infinite Lakshmibai-Seshadri (LS) paths and QLS paths. As an application, we give a crystal-theoretic proof of a recursion formula for the graded characters of generalized Weyl modules.
We introduce an affine Schur algebra via the affine Hecke algebra associated to Weyl group of affine type C. We establish multiplication formulas on the affine Hecke algebra and affine Schur algebra. Then we construct monomial bases and canonical bases for the affine Schur algebra. The multiplication formula allows us to establish a stabilization property of the family of affine Schur algebras that leads to the modified version of an algebra ${mathbf K}^{mathfrak c}_n$. We show that ${mathbf K}^{mathfrak c}_n$ is a coideal subalgebra of quantum affine algebra ${bf U}(hat{mathfrak{gl}}_n)$, and $big({mathbf U}(hat{ mathfrak{gl}}_n), {mathbf K}^{mathfrak c}_n)$ forms a quantum symmetric pair. The modified coideal subalgebra is shown to admit monomial and stably canonical bases. We also formulate several variants of the affine Schur algebra and the (modified) coideal subalgebra above, as well as their monomial and canonical bases. This work provides a new and algebraic approach which complements and sheds new light on our previous geometric approach on the subject. In the appendix by four of the authors, new length formulas for the Weyl groups of affine classical types are obtained in a symmetrized fashion.
We define the quantum group $D_4^+$ -- a free quantum version of the demihyperoctahedral group $D_4$ (the smallest representative of the Coxeter series $D$). In order to do so, we construct a free analogue of the property that a $4times4$ matrix has determinant one. Such analogues of determinants are usually very hard to define for free quantum groups in general and our result only holds for the matrix size $N=4$. The free $D_4^+$ is then defined by imposing this generalized determinant condition on the free hyperoctahedral group $H_4^+$. Moreover, we give a detailed combinatorial description of the representation category of $D_4^+$.
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