Related papers: On tensoring with the Steinberg representation

We study the Hodge filtrations of Schmid and Vilonen on unipotent representations of real reductive groups. We show that for various well-defined classes of unipotent representations (including, for example, the oscillator representations…

Let G be a semisimple algebraic group over an algebraically closed field of positive characteristic p. Generalizing the construction of the PBW filtration on Weyl modules for G we construct a G-stable filtration on tensor products of Weyl…

Let $\mathbf{G}$ be either a simple linear algebraic group over an algebraically closed field of characteristic $\ell>0$ or a quantum group at an $\ell$-th root of unity. We define a tensor ideal of singular $\mathbf{G}$-modules in the…

This paper consists of two prongs. Firstly, we prove that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a direct sum of graded simple modules. Secondly, we apply these…

Let $k$ be an algebraically closed field of characteristic $p$, possibly zero, and $G=q$-$\GL_3(k)$, the quantum group of three by three matrices as defined by Dipper and Donkin. We may also take $G$ to be $\GL_3(k)$. We first determine the…

We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $\mathcal{P}_\lambda$ of the quantum group $U_q(\mathfrak{sl}_{n+1})$ is closed under tensor products. Our…

In this paper we produce infinite families of counterexamples to Jantzen's question posed in 1980 on the existence of Weyl $p$-filtrations for Weyl modules for an algebraic group and Donkin's Tilting Module Conjecture formulated in 1990.…

We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree…

Let $n$ be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in $GL(n,{\mathbb{C}})/B$ such that its associated graded ring has graded…

Let $H$ be a finite dimensional pointed rank one Hopf algebra of nilpotent type. We first determine all finite dimensional indecomposable $H$-modules up to isomorphism, and then establish the Clebsch-Gordan formulas for the decompositions…

Let $G$ be a semisimple, simply connected algebraic group defined and split over a prime field ${\mathbb F}_p$ of positive characteristic. For a positive integer $r$, let $G_r$ be the $r$th Frobenius kernel of $G$. Let $Q$ be a projective…

Following the previous work, we determine exactly the highest weights for which a tensor product of two induced modules is a tilting module, for the algebraic group $SL_2$ over an algebraically closed field of positive characteristic.

Let $p$ be a prime number and $N$ an integer prime to $p$. We show that the operator $U_p$ on the space of cuspidal modular forms of level $pN$ and weight two is semi-simple. It follows from this that the Hecke algebra acting on the space…

We show the denominator formulas for the normalized $R$-matrix involving two arbitrary Kirillov--Reshetikhin (KR) modules $W^{(k)}_{m,a}$ and $W^{(l)}_{p,b}$ in all nonexceptional affine types, $D_4^{(3)}$, and $G_2^{(1)}$. To achieve our…

Given a smooth algebraic variety X with an action of a connected reductive linear algebraic group G, and an equivariant D-module M, we study the G-decompositions of the associated V-, Hodge, and weight filtrations. If M is the localization…

Let $K$ be a unramified $p$-adic field with the absolute Galois group $G_K$ and $T$ a crystalline $\mathbb Z_p$-representation of $G_K$. We study the graded pieces of integral filtration on $D_{\rm dR}(T)$ given by Nyggard filtration of the…

Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the ``order formula'' of D. Testerman…

Let G be a reductive algebraic group over a field k. When k=C, R.K.Brylinski constructed a filtration of weight spaces of a G module, using the action of a principal nilpotent element of the Lie algebra, and proved that this filtration…

The author and Nakano recently proved that multiplicities in a Specht filtration of a symmetric group module are well-defined precisely when the characteristic is at least five. This result suggested the possibility of a symmetric group…

Let $G$ be the group of rational points of a split connected reductive group over a nonarchimedean local field of residue characteristic $p$. Let $I$ be a pro-$p$ Iwahori subgroup of $G$ and let $R$ be a commutative quasi-Frobenius ring. If…