Related papers: Specht filtrations and tensor spaces for the Braue…

The Birkhoff polytope (the convex hull of the set of permutation matrices) is frequently invoked in formulating relaxations of optimization problems over permutations. The Birkhoff polytope is represented using $\Theta(n^2)$ variables and…

We consider the space $\text{Fun}_{\frak{sl}_2}V[0]$ of functions on the Cartan subalgebra of $\frak{sl}_2$ with values in the zero weight subspace $V[0]$ of a tensor product of irreducible finite-dimensional $\frak{sl}_2$-modules. We…

For each n>0, we define an algebra having many properties that one might expect to hold for a Brauer algebra of type Bn. It is defined by means of a presentation by generators and relations. We show that this algebra is a subalgebra of the…

We use the theory of skew duality to show that decomposing the tensor product of $k$ irreducible representations of the symplectic group $Sp_{2m} = Sp_{2m}(C)$ is equivalent to branching from $Sp_{2n}$ to $Sp_{2n_1}\times\cdots\times…

Let h \subset g be an inclusion of Lie algebras with quotient h-module n. There is a natural degree filtration on the h-module U(g)/U(g)h whose associated graded h-module is isomorphic to S(n). We give a necessary and sufficient condition…

Let $R$ be a commutative ring with one and $q$ an invertible element of $R$. The (specialized) quantum group ${\mathbf U} = U_q(\mathfrak{gl}_n)$ over $R$ of the general linear group acts on mixed tensor space $V^{\otimes r}\otimes…

We consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of degrees…

The affine Schur algebra $\widetilde{S}(n,r)$ (of type A) over a field $K$ is defined to be the endomorphism algebra of the tensor space over the extended affine Weyl group of type $A_{r-1}$. By the affine Schur-Weyl duality it is…

The Lawrence-Krammer representation was used in $2000$ to show the linearity of the braid group. The problem had remained open for many years. The fact that the Lawrence-Krammer representation of the braid group is reducible for some…

Using combinatorial properties of symmetric polynomials, we compute explicitly the Soergel modules for some permutations whose corresponding Schubert varieties are rationally smooth. We build from them diagram algebras whose module…

It is known that the recently discovered representations of the Artin groups of type A_n, the braid groups, can be constructed via BMW algebras. We introduce similar algebras of type D_n and E_n which also lead to the newly found faithful…

Let $M$ be a left module for the Schur algebra $S(n,r)$, and let $s \in \mathbb{Z}^+$. Then $M^{\otimes s}$ is a $(S(n,rs), F\mathfrak{S}_s)$-bimodule, where the symmetric group $\mathfrak{S}_s$ on $s$ letters acts on the right by place…

We introduce Brauer complex of symmetric SB-algebra, and reformulate in terms of Brauer complex the so far known invariants of stable and derived equivalence of symmetric SB-algebras. In particular, the genus of Brauer complex turns out to…

Let $n\geq 2$ be an integer, $S_n$ be the Lie algebra of vector fields on $\mathbb{C}^n$ with zero divergence, and $D_n$ be the Weyl algebra over the polynomial algebra $A_n=\mathbb{C}[t_1,t_2,\cdots,t_n]$. In this paper, we study the…

A construction of bases for cell modules of the Birman--Murakami--Wenzl (or B--M--W) algebra $B_n(q,r)$ by lifting bases for cell modules of $B_{n-1}(q,r)$ is given. By iterating this procedure, we produce cellular bases for B--M--W…

Given a complex orthosymplectic superspace $V$, the orthosymplectic Lie superalgebra $\mathfrak {osp}(V)$ and general linear algebra ${\mathfrak {gl}}_N$ both act naturally on the coordinate super-ring $\mathcal{S}(N)$ of the dual space of…

Using probabilistic tools, we prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of selfadjoint unital completely positive measurable Schur multipliers acting on the space $\mathrm{B}(\mathrm{L}^2(X))$ of bounded operators on the…

We define the affine VW supercategory $\mathit{s}\hspace{-0.7mm}\bigvee\mkern-15mu\bigvee$, which arises from studying the action of the periplectic Lie superalgebra $\mathfrak{p}(n)$ on the tensor product $M\otimes V^{\otimes a}$ of an…

We consider the action of the Lie algebra of polynomial vector fields, $\mathfrak{vect}(1)$, by the Lie derivative on the space of symbols $\mathcal{S}_\delta^n=\bigoplus_{j=0}^n \mathcal{F}_{\delta-j}$. We study deformations of this…

Motivated by the M-diagrams defined by Tymoczko, we show that these locally non-crossing $\mathfrak{sl}_3$-webs form a basis of the Specht module for the partition $(n,n,n)$. They further admit a unitriangular base change to both the…