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Register a new userEnumeration of torus-invariant strata with respect to dimension in the big cell of the quantum minuscule Grassmannian of type B_n
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The aim of this article is to give explicit formulae for various generating functions, including the generating function of torus-invariant primitive ideals in the big cell of the quantum minuscule grassmannian of type B_n.
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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 $Oq$.
The aim of this paper is to study the representation theory of quantum Schubert cells. Let $g$ be a simple complex Lie algebra. To each element $w$ of the Weyl group $W$ of $g$, De Concini, Kac and Procesi have attached a subalgebra $U_q[w]$ of the quantised enveloping algebra $U_q(g)$. Recently, Yakimov showed that these algebras can be interpreted as the quantum Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of $U_q[w]$. More precisely, it follows from the Stratification Theorem of Goodearl and Letzter that the primitive spectrum of $U_q[w]$ admits a stratification indexed by those primes that are invariant under a natural torus action. Moreover each stratum is homeomorphic to the spectrum of maximal ideals of a torus. The main result of this paper gives an explicit formula for the dimension of the stratum associated to a given torus-invariant prime.
This paper studies classical weight modules over the $imath$quantum group $mathbf{U}^{imath}$ of type AI. We introduce the notion of based $mathbf{U}^{imath}$-modules by generalizing the notion of based modules over the quantum groups. We prove that each finite-dimensional irreducible classical weight $mathbf{U}^{imath}$-module with integer highest weight is a based $mathbf{U}^{imath}$-module. As a byproduct, a new combinatorial formula for the branching rule from $mathfrak{sl}_n$ to $mathfrak{so}_n$ is obtained.
We use vertex operators to compute irreducible characters of the Iwahori-Hecke algebra of type $A$. Two general formulas are given for the irreducible characters in terms of those of the symmetric groups or the Iwahori-Hecke algebras in lower degrees. Explicit formulas are derived for the irreducible characters labeled by hooks and two-row partitions. Using duality, we also formulate a determinant type Murnaghan-Nakayama formula and give another proof of Rams combinatorial Murnaghan-Nakayama formula. As applications, we study super-characters of the Iwahori-Hecke algebra as well as the bitrace of the regular representation and provide a simple proof of the Halverson-Luduc-Ram formula.
We derive a formula for the trace of the antipode on endomorphism algebras of simple self-dual modules of nilpotent liftings of quantum planes. We show that the trace is equal to the quantum dimension of the module up to a nonzero scalar depending on the simple module.
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