Let $\mathfrak l:= \mathfrak q(n)\times\mathfrak q(n)$, where $\mathfrak q(n)$ denotes the queer Lie superalgebra. The associative superalgebra $V$ of type $Q(n)$ has a left and right action of $\mathfrak q(n)$, and hence is equipped with a canonical $\mathfrak l$-module structure. We consider a distinguished basis $\{D_λ\}$ of the algebra of $\mathfrak l$-invariant super-polynomial differential operators on $V$, which is indexed by strict partitions of length at most $n$. We show that the spectrum of the operator $D_λ$, when it acts on the algebra $\mathscr P(V)$ of super-polynomials on $V$, is given by the factorial Schur $Q$-function of Okounkov and Ivanov. This constitutes a refinement and a new proof of a result of Nazarov, who computed the top-degree homogeneous part of the Harish-Chandra image of $D_λ$. As a further application, we show that the radial projections of the spherical super-polynomials corresponding to the diagonal symmetric pair $(\mathfrak l,\mathfrak m)$, where $\mathfrak m:=\mathfrak q(n)$, of irreducible $\mathfrak l$-submodules of $\mathscr P(V)$ are the classical Schur $Q$-functions.
We show how the use of superalgebraic methods sheds new light on some classical themes of representation theory and it leads to significant simplifications of traditional proofs.
We define a natural basis for the algebra of $\frak{gosp}(1|2n)$-invariant differential operators on the affine superspace $\mathbb{C}^{1|2n}$. We prove that these operators lie in the image of the centre of the enveloping algebra of $\frak{gosp}(1|2n)$. Using this result, we compute explicit formulas for the eigenvalues of these operators on irreducible summands of $\mathcal{P}(\mathbb{C}^{1|2n})$. This settles the Capelli eigenvalue problem for orthosymplectic Lie superalgebras in the cases that were not addressed in Sahi-Salmasian-Serganova. Our main technique relies on an explicit calculat…
In this work, we propose a new method for a unified study of some of the main features of the theory of the center of the enveloping algebra U(gl(n)) and of the algebra of shifted symmetric polynomials, that allows the whole theory to be developed, in a transparent and concise way, from the representation-theoretic point of view, that is entirely in the center of U(gl(n)). Our methodological innovation is the systematic use of the superalgebraic method of virtual variables for gl(n), which is, in turn, an extension of Capelli's method of "variabili ausiliarie".
For a finite dimensional unital complex simple Jordan superalgebra $J$, the Tits-Kantor-Koecher construction yields a 3-graded Lie superalgebra $\mathfrak g_\flat\cong \mathfrak g_\flat(-1)\oplus\mathfrak g_\flat(0)\oplus\mathfrak g_\flat(1)$, such that $\mathfrak g_\flat(-1)\cong J$. Set $V:=\mathfrak g_\flat(-1)^*$ and $\mathfrak g:=\mathfrak g_\flat(0)$. In most cases, the space $\mathcal P(V)$ of superpolynomials on $V$ is a completely reducible and multiplicity-free representation of $\mathfrak g$, with a decomposition $\mathcal P(V):=\bigoplus_{λ\inΩ}V_λ$, where $\left(V_λ\right)_{λ\inΩ}…
In this paper are introduced two classes of elements in the enveloping algebra $\mathbf{U}(gl(n))$: the \emph{double Young-Capelli bitableaux} $[\ \fbox{$S \ | \ T$}\ ]$ and the \emph{central} \emph{Schur elements} $\mathbf{S}_λ(n)$, that act in a remarkable way on the highest weight vectors of irreducible Schur modules. Any element $\mathbf{S}_λ(n)$ is the sum of all double Young-Capelli bitableaux $[\ \fbox{$S \ | \ S$}\ ]$, $S$ row (strictly) increasing Young tableaux of shape $\widetildeλ$. The Schur elements $\mathbf{S}_λ(n)$ are proved to be the preimages - with respect to the Harish-Cha…
The "Capelli problem" for the symmetric pairs $(\mathfrak{gl}\times \mathfrak{gl},\mathfrak{gl})$ $(\mathfrak{gl},\mathfrak{o})$, and $(\mathfrak{gl},\mathfrak{sp})$ is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter. In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs $(\mathfrak{g},\mathfrak{g}):=(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$ and $(\mathfrak{gl}(m|n)\times\mathfrak{gl}(m|n),\mathfrak{gl}(m|n))$, acting on $W:=S^2(\mathbb C^{m|2n})$ and $\mathbb C^{m|n}\oti…
We provide a presentation of the Schur superalgebra and its quantum analogue which generalizes the work of Doty and Giaquinto for Schur algebras. Our results include a basis for these algebras and a presentation using weight idempotents in the spirit of Lusztig's modified quantum groups.
Let $S(m|n,d)$ be the Schur superalgebra whose supermodules correspond to the polynomial representations of the supergroup $GL(m|n)$ of degree $d$. In this paper we determine the representation type of these algebras (i.e. classify the ones which are semisimple, have finite, tame and wild representation type). Moreover, we prove that these algebras are in general not quasi-hereditary and have infinite global dimension.
Let $\mathfrak g$ be either the Lie superalgebra $\mathfrak{gl}(V)\oplus\mathfrak{gl}(V)$ where $V:=\mathbb C^{m|n}$ or the Lie superalgebra $\mathfrak{gl}(V)$ where $V:=\mathbb C^{m|2n}$. Furthermore, let $W$ be the $\mathfrak g$-module defined by $W:=V\otimes V^*$ in the former case and $W:=\mathcal S^2(V)$ in the latter case. Associated to $(\mathfrak g,W)$ there exists a distinguished basis of Capelli operators $\left\{D^λ\right\}_{λ\inΩ}$, naturally indexed by a set of hook partitions $Ω$, for the subalgebra of $\mathfrak g$-invariants in the superalgebra $\mathcal{PD}(W)$ of superdiffere…
Making use of a Howe duality involving the infinite-dimensional Lie superalgebra $\hgltwo$ and the finite-dimensional group $GL_l$ we derive a character formula for a certain class of irreducible quasi-finite representations of $\hgltwo$ in terms of hook Schur functions. We use the reduction procedure of $\hgltwo$ to $\hat{gl}_{n|n}$ to derive a character formula for a certain class of level 1 highest weight irreducible representations of $\hat{gl}_{n|n}$, the affine Lie superalgebra associated to the finite-dimensional Lie superalgebra $gl_{n|n}$. These modules turn out to form the complete s…
We introduce the notion of $\imath$Schur superalgebra, which can be regarded as a type B/C counterpart of the $q$-Schur superalgebra (of type A) formulated as centralizer algebras of certain signed $q$-permutation modules over Hecke algebras. Some multiplication formulas for $\imath$Schur superalgebra are obtained to construct their canonical bases. Furthermore, we established an isomorphism theorem between the $\imath$Scuhr superalgebras and the $q$-Schur superalgebras of type A, which helps us derive a semisimplicity criteria of the $\imath$Schur superalgebras.
The paper aims to introduce the cyclotomic $q$-Schur superalgebra via the permutation supermodules of the cyclotomic Hekce algebra and investigate its structure. In particular, we show that the cyclotomic $q$-Schur superalgebra is a cellular superalgebra and establish the Schur-Weyl duality between the cyclotomic Hecke algebra and the cyclotomic $q$-Schur superalgebra.
Let A_k denote the twisted group algebra of the symmetric group S_k, whose representations correspond to the nonlinear projective representations of S_k. We establish a duality relation between A_k and a Lie superalgebra q(n), sometimes called the ``queer'' Lie superalgebra. This duality clarifies Schur's formula, in the sense that the Schur-Weyl duality clarifies Frobenius' formula.
Two super-analogs of the Schur-Weyl duality are considered: the duality of actions in $(\mathbb{C}^{m|n})^{\otimes N}$ of the Lie superalgebra $\mathfrak{gl}(m,n)$ and the symmetric group $S_N$, and the duality of actions of the Lie superalgebra $Q(n)$ and a certain finite group $Se(N)$ in $(\mathbb{C}^{n|n})^{\otimes N}$. We construct an isomorphism of symmetric and universal enveloping algebras of these Lie superalgebras called special symmetrization. Using this isomorphism of vector spaces we describe explicitly the duality between the centers of the corresponding universal enveloping algeb…
Given a quasi-hereditary superalgebra $A$, the first author and R. Muth have defined generalized Schur bi-superalgebras $T^A(n)$ and proved that these algebras are again quasi-hereditary. In particular, $T^A(n)$ comes with a family of standard modules. Developing the work of Donkin and Mathieu on good filtrations, we prove that tensor product of standard modules over $T^A(n)$ has a standard filtration.
We introduce a class of Schur type functions associated with polynomial sequences of binomial type. This can be regarded as a generalization of the ordinary Schur functions and the factorial Schur functions. This generalization satisfies some interesting expansion formulas, in which there is a curious duality. Moreover this class includes examples which are useful to describe the eigenvalues of Capelli type central elements of the universal enveloping algebras of classical Lie algebras.
Associated to the two types of finite dimensional simple superalgebras, there are the general linear Lie superalgebra and the queer Lie superalgebra. The universal enveloping algebras of these Lie superalgebras act on the tensor spaces of the natural representations and, thus, define certain finite dimensional quotients, the Schur superalgebras and the queer Schur superalgebra. In this paper, we introduce the quantum analogue of the queer Schur superalgebra and investigate the presentation problem for both the queer Schur superalgebra and its quantum analogue.
Let $G=GL(m|n)$ be the general linear supergroup over an algebraically closed field $K$ of characteristic zero and let $G_{ev}=GL(m)\times GL(n)$ be its even subsupergroup. The induced supermodule $H^0_G(λ)$, corresponding to a dominant weight $λ$ of $G$, can be represented as $H^0_{G_{ev}}(λ)\otimes Λ(Y)$, where $Y=V_m^*\otimes V_n$ is a tensor product of the dual of the natural $GL(m)$-module $V_m$ and the natural $GL(n)$-module $V_n$, and $Λ(Y)$ is the exterior algebra of $Y$. For a dominant weight $λ$ of $G$, we construct explicit $G_{ev}$-primitive vectors in $H^0_G(λ)$. Related to this, …
In this paper, we introduce the sign q-permutation representation of the Iwahori-Hecke algebra on the tensor space of the graded vector space. We establish Schur-Weyl reciprocity between the quantum general super Lie algebra and the Iwahori-Hecke algebra. Our result includes Schur-Weyl reciprocity of both the super case and the quantum case, and unifies those two important cases.
As a natural generalization quantum Schur algebras associated with the Hecke algebra of the symmetric group, we introduce the quantum Schur superalgebra of type Q associated with the Hecke-Clifford superalgebra, which, by definition, is the endomorphism algebra of the induced module over the Hecke-Clifford superalgebra from certain permutation modules over the Hecke algebra of the symmetric group. We will describe certain integral bases for these superalgebras in terms of matrices and will establish the base change property for them. We will also identify the quantum Schur superalgebra of type…
Degenerating the quantum queer Schur superalgebra ${\mathcal{Q}_q(n,r; R)}$ to the case $q=1$, the queer Schur superalgebra ${\mathcal{Q}(n,r)}$ is obtained. In this article, we reconstruct the universal enveloping algebra ${U({\mathfrak{q}_n})}$ of the queer Lie superalgebra ${\mathfrak{q}_n}$ via ${\mathcal{Q}(n,r)}$, and achieve another explanation of the Schur-Weyl-Sergeev duality. Finally, we depict the Schur-Weyl-Sergeev duality over $\mathbb{Z}$.
We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for q-Schur superalgebras. This gives a combinatorialisation of the relative norm approach developed by the first two authors. We then give several applications of the multiplication formulas, including the matrix representation of the regular representation and a semisimplicity criterion for q-Schur superalgebras. We also construct infinitesimal and little q-Schur superalgebras directly from the multiplication formulas and develop their semisimplicity criteria.
We introduce categories of homogeneous strict polynomial functors, $\Pol^\I_{d,\k}$ and $\Pol^\II_{d,\k}$, defined on vector superspaces over a field $\k$ of characteristic not equal 2. These categories are related to polynomial representations of the supergroups $GL(m|n)$ and Q(n), respectively. In particular, we prove an equivalence between $\Pol^\I_{d,\k}$, $\Pol^\II_{d,\k}$ and the category of finite dimensional supermodules over the Schur superalgebra $\Sc(m|n,d)$, $\Qc(n,d)$ respectively provided $m,n \ge d$. We also discuss some aspects of Sergeev duality from the viewpoint of the categ…
Under the assumption that the quantum parameter $q$ is an $l$-th primitive root of unity with $l$ odd in a field $F$ of characteristic 0 and $m+n\geq r$, we obtained a complete classification of irreducible modules of the $q$-Schur superalgebra introduced H. Rui and the first Author.
We introduce a new family of superalgebras $\overrightarrow{B}_{r,s}$ for $r, s \ge 0$ such that $r+s>0$, which we call the walled Brauer superalgebras, and prove the mixed Scur-Weyl-Sergeev duality for queer Lie superalgebras. More precisely, let $\mathfrak{q}(n)$ be the queer Lie superalgebra, ${\mathbf V} =\mathbb{C}^{n|n}$ the natural representation of $\mathfrak{q}(n)$ and ${\mathbf W}$ the dual of ${\mathbf V}$. We prove that, if $n \ge r+s$, the superalgebra $\overrightarrow{B}_{r,s}$ is isomorphic to the supercentralizer algebra $_{\mathfrak{q}(n)}({\mathbf V}^{\otimes r} \otimes {\mat…
Let $(V,ω)$ be an orthosympectic $\mathbb Z_2$-graded vector space and let $\mathfrak g:=\mathfrak{gosp}(V,ω)$ denote the Lie superalgebra of similitudes of $(V,ω)$. When the space $\mathscr P(V)$ of superpolynomials on $V$ is \emph{not} a completely reducible $\mathfrak g$-module, we construct a natural basis $D_λ$ of Capelli operators for the algebra of $\mathfrak g$-invariant superpolynomial superdifferential operators on $V$, where the index set $\mathcal P$ is the set of integer partitions of length at most two. We compute the action of the operators $D_λ$ on maximal indecomposable compon…
We introduce super-analogues of the Schur functors defined by Akin, Buchsbaum and Weyman. These {\em Schur superfunctors} may be viewed as characteristic-free analogues of the finite dimensional atypical irreducible modules over the Lie superalgebra $\mathfrak{gl}(m|n)$ studied by Berele and Regev. Our construction realizes Schur superfunctors as objects of a certain category of strict polynomial superfunctors. We show that Schur superfunctors are indecomposable objects of this category. Another aim is to provide a decomposition of {\em Schur bisuperfunctors} in terms of tensor products of Sch…
The paper contains results that characterize the Donkin-Koppinen filtration of the coordinate superalgebra $K[G]$ of the general linear supergroup $G=GL(m|n)$ by its subsupermodules $C_Γ=O_Γ(K[G])$. Here, the supermodule $C_Γ$ is the largest subsupermodule of $K[G]$ whose composition factors are irreducible supermodules of highest weight $λ$, where $λ$ belongs to a finitely-generated ideal $Γ$ of the poset $X(T)^+$ of dominant weights of $G$. A decomposition of $G$ as a product of subsuperschemes $U^-\times G_{ev}\times U^+$ induces a superalgebra isomorphism $ϕ^* : K[U^-]\otimes K[G_{ev}]\oti…
I show that the projective Schur functions may be interpreted as bispherical functions of either the triple (q(n),gl(n,n),q(n)), where q(n) is the "odd" (queer) analog of the general liner Lie algebra, or the triple (p(n),gl(n,n),p(n)), where p(n) is the periplectic Lie superalgebra which preserves the nondegenerate odd, bilinear form (either symmetric or skew symmetric). Making use of this interpretation I characterize projective Schur functions as common eigenfunctions of an algebra differential operators.
We define and study new classes of quasi-hereditary and cellular algebras which generalize Turner's double algebras. Turner's algebras provide a local description of blocks of symmetric groups up to derived equivalence. Our general construction allows one to `schurify' any quasi-hereditary algebra $A$ to obtain a generalized Schur algebra $S^A(n,d)$ which we prove is again quasi-hereditary if $d\leq n$. We describe decomposition numbers of $S^A(n,d)$ in terms of those of $A$ and the classical Schur algebra $S(n,d)$. In fact, it is essential to work with quasi-hereditary superalgebras $A$, in w…
In this master's thesis, we recall the definitions and basic results for Lie superalgebras. We specify the definition for Klein graded Lie algebras and, motivated by well known results for Lie superalgebras, we prove similar results for Klein graded Lie algebras. More precisely, we state the theorems of Poincare-Birkhoff-Witt and Ado's, as well as Schur's lemma. Moreover, we present two examples of Klein graded algebra.
For the Schur superalgebra $S=S(m|n,r)$ over a ground field $K$ of characteristic zero, we define symmetrizers $T^λ[i:j]$ of the ordered pairs of tableaux $T_i, T_j$ of the shape $λ$ and show that the $K$-span $A_{λ,K}$ of all symmetrizers $T^λ[i:j]$ has a basis consisting of $T^λ[i:j]$ for $T_i,T_j$ semistandard. The $S$-superbimodule $A_{λ,K}$ is identified as %$Δ(λ)^*\otimes_K \nabla(λ)$, where $Δ(λ)^*$ is the dual of the standard supermodule %and $\nabla(λ)$ is the costandard supermodule of the highest weight $λ$. $D_λ\otimes_K D^o_λ$, where $D_λ$ and $D^o_λ$ are left and right irreducible…
Let $\mathbb{k}$ be a characteristic zero domain. For a locally unital $\mathbb{k}$-superalgebra $A$ with distinguished idempotents $I$and even subalgebra $a \subseteq A_{\bar 0}$, we define and study an associated diagrammatic monoidal $\mathbb{k}$-linear supercategory $\mathbf{Web}^{A,a}_I$. This supercategory yields a diagrammatic description of the generalized Schur algebras $T^A_a(n,d)$. We also show there is an asymptotically faithful functor from $\mathbf{Web}^{A,a}_I$ to the monoidal supercategory of $\mathfrak{gl}_n(A)$-modules generated by symmetric powers of the natural module. When…
In this note, we investigate a kind of double centralizer property for general linear supergroups. For the super space $V=\mathbb{K}^{m\mid n}$ over an algebraically closed field $\mathbb{K}$ whose characteristic is not equal to $2$, we consider its $\mathbb{Z}_2$-homogeneous one-dimensional extension $\underline V=V\oplus\mathbb{K}v$, and the natural action of the supergroup $\tilde G:=\text{GL}(V)\times \textbf{G}_m$ on $\underline V$. Then we have the tensor product supermodule ($\underline{V}^{\otimes r}$, $ρ_r$) of $\tilde G$. We present a kind of generalized Schur-Sergeev duality which i…
In this paper, we present a simple combinatorial proof of a Weyl type formula for hook Schur polynomials, which has been obtained by using a Kostant type cohomology formula for $\frak{gl}_{m|n}$. In general, we can obtain in a combinatorial way a Weyl type formula for various highest weight representations of a Lie superalgebra, which together with a general linear algebra forms a Howe dual pair.
We reconstruct the quantum enveloping superalgebra ${\bf U}(\mathfrak{gl}_{m|n})$ over $\mathbb Q(v)$ via (finite dimensional) quantum Schur superalgebras. In particular, we obtain a new basis containing the standard generators of ${\bf U}(\mathfrak{gl}_{m|n})$ and explicit multiplication formulas between the generators and an arbitrary basis element.
We explain how Lie superalgebras of types gl and osp provide a natural framework generalizing the classical Schur and Howe dualities. This exposition includes a discussion of super duality, which connects the parabolic categories O between classical Lie superalgebras and Lie algebras. Super duality provides a conceptual solution to the irreducible character problem for these Lie superalgebras in terms of the classical Kazhdan-Lusztig polynomials.
We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras $\mathfrak{p}(n)$ as $n \to \infty$. The paper gives a construction of the tensor category $Rep(\underline{P})$, possessing nice universal properties among tensor categories over the category $\mathtt{sVect}$ of finite-dimensional complex vector superspaces. First, it is the "abelian envelope" of the Deligne category corresponding to the periplectic Lie superalgebra, in the sense of arXiv:1511.07699. Secondly, given a tensor category $\mathcal{C}$ over $\mathtt{sVect}$, exact tensor functors $Rep(…
We prove that Euler supercharacters for orthosymplectic Lie superalgebras can be obtained as a certain specialization of super Jacobi polynomials. A new version of Weyl type formula for super Schur functions and specialized super Jacobi polynomials play a key role in the proof.
Let $U_q(\mathfrak{g})$ be the quantized superalgebra of $\mathfrak{g}=\mathfrak{gl}(k_1|\ell_1)\oplus\cdots\oplus\mathfrak{gl}(k_m|\ell_m)$ and $H_{m,n}(q,\mathbf{Q})$ the cyclotomic Hecke algebra of type $G(m,1,n)$. We define a right $H_{m,n}(q,\mathbf{Q})$-action on the $n$-fold tensor (super)space of the vector representation of $U_q(\mathfrak{g})$ and prove the Schur--Weyl reciprocity between $U_q(\mathfak{g})$ and $H_{m,n}(q,\mathbf{Q})$.
In this paper, a notion of cyclotomic (or level $k$) walled Brauer algebras $\mathscr B_{k, r, t}$ is introduced for arbitrary positive integer $k$. It is proven that $\mathscr B_{k, r, t}$ is free over a commutative ring with rank $k^{r+t}(r+t)!$ if and only if it is admissible. Using super Schur-Weyl duality between general linear Lie superalgebras $\mathfrak{gl}_{m|n}$ and $\mathscr B_{2, r, t}$, we give a classification of highest weight vectors of $\mathfrak{gl}_{m|n}$-modules $M_{pq}^{rt}$, the tensor products of Kac-modules with mixed tensor products of the natural module and its dual. …
We introduce the marked Brauer algebra and the marked Brauer category. These generalize the analogous constructions for the ordinary Brauer algebra to the setting of a homogeneous bilinear form on a $\mathbb{Z}_2$-graded vector space. We classify the simple modules of the marked Brauer algebra over any field of characteristic not two. Under suitable assumptions we show that the marked Brauer algebra is in Schur-Weyl duality with the Lie superalgebra, $\mathfrak{g}$, of linear maps which leave the bilinear form invariant. We also provide a classification of the indecomposable summands of the te…
