Algebras and Representation Theory最新文献

Let (mathfrak {g}) be a complex semisimple Lie algebra. We give a description of characters of irreducible Whittaker modules for (mathfrak {g}) with any infinitesimal character, along with a Kazhdan-Lusztig algorithm for computing them. This generalizes Miličić-Soergel’s and Romanov’s results for integral infinitesimal characters. As a special case, we recover the non-integral Kazhdan-Lusztig conjecture for Verma modules.

We study the tropical dualities and properties of exchange graphs for the totally sign-skew-symmetric cluster algebra under a condition. We prove that the condition always holds for acyclic cluster algebras, then all results hold for the acyclic case.

In this article the quantized matrix algebras as in the title have been studied at a root of unity. A full classification of simple modules over such quantized matrix algebras of degree 2 along with some finite-dimensional indecomposable modules are explicitly presented.

To each triangulation of any surface with marked points on the boundary and orbifold points of order three, we associate a quiver (with loops) with potential whose Jacobian algebra is finite dimensional and gentle. We study the stability scattering diagrams of such gentle algebras and use them to prove that the Caldero–Chapoton map defines a bijection between (tau )-rigid pairs and cluster monomials of the generalized cluster algebra associated to the surface by Chekhov and Shapiro.

We consider the dominant dimension of an order over a Cohen-Macaulay ring in the category of centrally Cohen-Macaulay modules. There is a canonical tilting module in the case of positive dominant dimension and we give an upper bound on the global dimension of its endomorphism ring.

In this paper, we associate a finite dimensional algebra, called a Brauer graph algebra, to every clean dessin d’enfant by constructing a quiver based on the monodromy of the dessin. We show that Galois conjugate dessins d’enfants give rise to derived equivalent Brauer graph algebras and that the stable Auslander-Reiten quiver and the dimension of the Brauer graph algebra are invariant under the induced action of the absolute Galois group.

The aim of this article is to consider the spectral sequences induced by tensor-hom adjunction, and provide a number of new results. Let R be a commutative Noetherian local ring of dimension d. In the 1st part, it is proved that R is Gorenstein if and only if it admits a nonzero CM (Cohen-Macaulay) module M of finite Gorenstein dimension g such that (text {type}(M) leqslant mu ( text {Ext}_R^g(M,R) )) (e.g., (text {type}(M)=1)). This considerably strengthens a result of Takahashi. Moreover, we show that if there is a nonzero R-module M of depth (geqslant d - 1) such that the injective dimensions of M, (text {Hom}_R(M,M)) and (text {Ext}_R^1(M,M)) are finite, then M has finite projective dimension and R is Gorenstein. In the 2nd part, we assume that R is CM with a canonical module (omega ). For CM R-modules M and N, we show that the vanishing of one of the following implies the same for others: (text {Ext}_R^{gg 0}(M,N^{+})), (text {Ext}_R^{gg 0}(N,M^{+})) and (text {Tor}_{gg 0}^R(M,N)), where (M^{+}) denotes (text {Ext}_R^{d-dim (M)}(M,omega )). This strengthens a result of Huneke and Jorgensen. Furthermore, we prove a similar result for Tate cohomologies under the additional condition that R is Gorenstein.

Demazure crystals are subcrystals of highest weight irreducible (mathfrak {g})-crystals. In this article, we study tensor products of a larger class of subcrystals, called extremal, and give a local characterization for exactly when the tensor product of Demazure crystals is extremal. We then show that tensor products of Demazure crystals decompose into direct sums of Demazure crystals if and only if the tensor product is extremal, thus providing a sufficient and necessary local criterion for when the tensor product of Demazure crystals is itself Demazure. As an application, we show that the primary component in the tensor square of any Demazure crystal is always Demazure.

In terms of local cohomology, we give an explicit range as to when the FI-homology of an FI-module attains the degree predicted by its regularity.

We compute the auto-correlations functions of order (mge 1) for the characteristic polynomials of random matrices from certain subgroups of the unitary groups ({text {U}}(2)) and ({text {U}}(3)) by establishing new branching rules. These subgroups can be understood as certain analogues of Sato–Tate groups of ({text {USp}}(4)) in our previous paper. Our computation yields symmetric polynomial identities with m-variables involving irreducible characters of ({text {U}}(m)) for all (m ge 1) in an explicit, uniform way.