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Register a new userPrimitive ideals in quantum Schubert cells: dimension of the strata
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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.
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Let ${mathfrak p}subset {mathfrak g}$ be a parabolic subalgebra of s simple finite dimensional Lie algebra over ${mathbb C}$. To each pair $w^{mathfrak a}leq w^{mathfrak c}$ of minimal left coset representatives in the quotient space $W_pbackslash W$ we construct explicitly a quantum seed ${mathcal Q}_q({mathfrak a},{mathfrak c})$. We define Schubert creation and annihilation mutations and show that our seeds are related by such mutations. We also introduce more elaborate seeds to accommodate our mutations. The quantized Schubert Cell decomposition of the quantized generalized flag manifold can be viewed as the result of such mutations having their origins in the pair $({mathfrak a},{mathfrak c})= ({mathfrak e},{mathfrak p})$, where the empty string ${mathfrak e}$ corresponds to the neutral element. This makes it possible to give simple proofs by induction. We exemplify this in three directions: Prime ideals, upper cluster algebras, and the diagonal of a quantized minor.
The goal of this work is to provide an elementary construction of the canonical basis $mathbf B(w)$ in each quantum Schubert cell~$U_q(w)$ and to establish its invariance under modified Lusztigs symmetries. To that effect, we obtain a direct characterization of the upper global basis $mathbf B^{up}$ in terms of a suitable bilinear form and show that $mathbf B(w)$ is contained in $mathbf B^{up}$ and its large part is preserved by modified Lusztigs symmetries.
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 discuss a relationship between Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds, Fomin-Kirillov algebra, and the generalized nil-Hecke algebra. We show that nonnegativity conjecture in Fomin-Kirillov algebra implies the nonnegativity of the Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of the Bott-Samelson varieties. We also discuss the refined positivity conjectures of the Chern-Schwartz-MacPherson classes for Schubert cells motivated by the nonnegativity conjecture in Fomin-Kirillov algebra.
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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