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We state and prove an analog of the Schur-Weyl duality for a quotient of the classical 2-toroidal Lie algebra of type A. We then provide a method to extend this duality to the m-toroidal case, m > 2. [ABSTRACT FROM AUTHOR]

The quasi-partition algebras were introduced in [10] as centralizers of the symmetric group. In this article, we give a more general definition of these algebras and give a construction of their simple modules. In addition, we introduce two new algebras, we give linear bases and show that for specializations of their parameters, these new algebras are isomorphic to centralizer algebras. We provide a generalized Bratteli diagram that illustrates how the representation theory of the three algebras discussed in this paper are related. Moreover, we give combinatorial formulas for the dimensions of the simple modules of these algebras. [ABSTRACT FROM AUTHOR]

Under a suitable hypothesis, we construct a full set of pairwise orthogonal maximal vectors in V ⊗ n , where V = V (1) is the simple module of highest weight 1 for the quantized enveloping algebra U (sl 2). We give a number of applications, one of which is an orthogonal basis of the simple modules for the Temperley–Lieb algebra TL n. We relate this new orthogonal basis to the standard cellular basis. [ABSTRACT FROM AUTHOR]

We propose a new approach to study plethysm coefficients by using the Schur-Weyl duality between the symmetric group and the partition algebra. This provides an explanation of the stability properties of plethysm and Kronecker coefficients in a simple and uniform fashion for the first time. We prove the strengthened Foulkes' conjecture for stable plethysm coefficients in an elementary fashion. [ABSTRACT FROM AUTHOR]

The rook monoid, also known as the symmetric inverse monoid, is the archetypal structure when it comes to extend the principle of symmetry. In this paper, we establish a Schur–Weyl duality between this monoid and an extension of the classical Schur algebra, which we name the extended Schur algebra. We also explain how this relates to Solomon's Schur–Weyl duality between the rook monoid and the general linear group and mention some advantages of our approach. [ABSTRACT FROM AUTHOR]

We introduce a certain discrete probability distribution P n , m , k , l ; q having non-negative integer parameters n, m, k, l and quantum parameter q which arises from a zonal spherical function of the Grassmannian over the finite field F q with a distinguished spherical vector. Using representation theoretic arguments and hypergeometric summation technique, we derive the presentation of the probability mass function by a single q-Racah polynomial, and also the presentation of the cumulative distribution function in terms of a terminating 4 ϕ 3 -hypergeometric series. [ABSTRACT FROM AUTHOR]

In this paper, we define a G -graded Hopf module associative pseudoalgebra A for any grading group G and Hopf algebra H , and construct a smash product A # H of A with H , which is also a G -graded associative pseudoalgebra. Then we establish a Mortia context between A # H and A H , the fixed sub-pseudoalgebra under the action of H. In addition, we prove a similarity of the Schur-Weyl type duality between the group algebra k [ S e (n) ] of Sergeev group S e (n) and a super pseudoalgebra. [ABSTRACT FROM AUTHOR]

We define the notion of representation stability for sequences of modules over the Temperley–Lieb algebras, the Brauer algebras, and the partition algebras in the semisimple setting analogous to Church and Farb. We also find a condition when such sequences are representation stable analogous to Church, Ellenberg, and Farb. To this end, we introduce stability categories for these diagram algebras—analogs to Randal-Williams and Wahl's homogeneous categories. We prove that finitely presented modules over these stability categories give rise to representation stable sequences as long as the parameters ensure semisimplicity. [ABSTRACT FROM AUTHOR]

We study the symplectic Howe duality using two new and independent combinatorial methods: via determinantal formulae on the one hand, and via (bi)crystals on the other hand. The first approach allows us to establish a generalised version where weight multiplicities are replaced by branching coefficients. In turn, this generalised Howe duality is used to prove the injectivity of induction for Levi branchings as previously conjectured by the last two authors.Mathematics Subject Classifications: 17B10, 17B37, 05E05, 05E10Keywords: Lie algebras, representation theory, Schur-Weyl duality, Howe duality, crystals, Schur functions, induced modules

We define an affine partition algebra by generators and relations and prove a variety of basic results regarding this new algebra analogous to those of other affine diagram algebras. In particular we show that it extends the Schur-Weyl duality between the symmetric group and the partition algebra. We also relate it to the affine partition category recently defined by J. Brundan and M. Vargas. Moreover, we show that this affine partition category is a full monoidal subcategory of the Heisenberg category. [ABSTRACT FROM AUTHOR]

The affine Temperley–Lieb algebra aTLN(β) is an infinite-dimensional algebra over ℂ parametrized by a number β ∈ ℂ and an integer N ∈ ℕ . It naturally acts on (ℂ 2) ⊗ N to produce a family of representations labeled by an additional parameter z ∈ ℂ × . The structure of these representations, which were first introduced by Pasquier and Saleur (Nucl. Phys., 330, 523 1990) in their study of spin chains, is here made explicit. They share their composition factors with the cellular aTLN(β)-modules of Graham and Lehrer (Enseign. Math., 44, 173 1998), but differ from the latter by the direction of about half of the arrows of their Loewy diagrams. The proof of this statement uses a morphism introduced by Morin-Duchesne and Saint-Aubin (J. Phys. A, 46, 285207 2013) as well as new maps that intertwine various aTLN(β)-actions on the periodic chain and generalize applications studied by Deguchi et al. (J. Stat. Phys., 102, 701 2001) and after by Morin-Duchesne and Saint-Aubin (J. Phys. A, 46, 494013 2013). [ABSTRACT FROM AUTHOR]

Let V be a Z -graded vertex operator superalgebra and G a finite automorphism group of V of order T. For g ∈ G and n ∈ 1 T Z + , a series of associative algebras A g , n (V) are constructed to study the twisted representations of V. In addition, we introduce a G -twisted associative algebra A G , n (V) such that A g , n (V) is a quotient algebra of A G , n (V) for any g ∈ G. As an application of the theory of A G , n (V) , we obtain a duality theorem of Schur-Weyl type. In particular, for any g ∈ G , every irreducible g -twisted V-module is a completely reducible V G -module. [ABSTRACT FROM AUTHOR]

Let K be a field, Γ a finite group of field automorphisms of K, F the Γ -fixed field in K, and G ≤ GL v (K) a finite matrix group. Then the action of Γ defines a grading on the symmetric algebra of the F-space K v which we use to introduce the notion of homogeneous Γ -conjugate invariants of G. We apply this new grading in invariant theory to broaden the connection between codes and invariant theory by introducing Γ -conjugate complete weight enumerators of codes. The main result of this paper applies the theory from Nebe, Rains, Sloane to show that under certain extra conditions these new weight enumerators generate the ring of Γ -conjugate invariants of the associated Clifford–Weil groups. As an immediate consequence, we obtain a result by Bannai et al. that the complex conjugate weight enumerators generate the ring of complex conjugate invariants of the complex Clifford group. Also the Schur–Weyl duality conjectured and partly shown by Gross et al. can be derived from our main result. [ABSTRACT FROM AUTHOR]

It is a well-established fact in quantum information theory, that uniform averaging over the collective action of a unitary group on a multipartite quantum state projects the state to a form equivalent to a permutation operator of the subsystems. Hence states equivalent to permutation operators are untouched by collective unitary noise. A trivial observation shows that uniform averaging over permutation operators projects the state into a form with block-diagonal structure equivalent to the one of the collective action of the unitary group. We introduce a name for this property: duality of averaging. The mathematical reason behind this duality is the fact that the collective action of the unitary group on the tensor product state space of a multipartite quantum system and the action of the permutation operations are mutual commutants when treated as matrix algebras. Such pairs of matrix algebras are known as dual reductive pairs. In this work we show, that in the case of finite dimensional quantum systems such duality of averaging holds for any pairs of symmetry groups being dual reductive pairs, regardless of whether they are compact or not, as long as the averaging operation is defined via iterated integral over the Cartan decomposition of the group action. Although our result is very general, we focus much attention on the concrete example of a dual reductive pair consisting of collective action of special linear matrices and permutation operations, which physically corresponds to averaging multipartite quantum states over non-unitary SLOCC-type (Stochastic Local Operations and Classical Communication) operations. In this context we show, that noiseless subsystems known from collective unitary averaging persist in the case of SLOCC averaging in a conditional way: upon postselection to specific invariant subspaces. [ABSTRACT FROM AUTHOR]

In this paper, we give a combinatorial description of a new diagram algebra, the partial Temperley-Lieb algebra, arising as the generic centralizer algebra EndUq(gl2)(V⊕k/, where V = V(0) ⊕ V(1) is the direct sum of the trivial and natural module for the quantized enveloping algebra Uq(gl2). It is a proper subalgebra of the Motzkin algebra (the Uq(sl2)-centralizer) of Benkart and Halverson. We prove a version of Schur-Weyl duality for the new algebras, and describe their generic representation theory. [ABSTRACT FROM AUTHOR]

We present in this paper the algebra of fused permutations and its deformation the fused Hecke algebra. The first one is defined on a set of combinatorial objects that we call fused permutations, and its deformation is defined on a set of topological objects that we call fused braids. We use these algebras to prove a Schur–Weyl duality theorem for any tensor products of any symmetrised powers of the natural representation of Uq(glN). Then we proceed to the study of the fused Hecke algebras and in particular, we describe explicitly the irreducible representations and the branching rules. Finally, we aim to give an algebraic description of the centralisers of the tensor products of Uq(glN)-representations under consideration. We exhibit a simple explicit element that we conjecture to generate the kernel from the fused Hecke algebra to the centraliser. We prove this conjecture in some cases and in particular, we obtain a description of the centraliser of any tensor products of any finite-dimensional representations of Uq(sl2). [ABSTRACT FROM AUTHOR]

In this paper, we generalize Schur–Weyl duality and Morita Theorem on associative algebras to those on associative H-pseudoalgebras. Meanwhile, we get a plenty of associative H-pseudoalgebras over a cocommutative Hopf algebra H [ABSTRACT FROM AUTHOR]

We consider the problem of determining the achievable region of the universal 1 → 2 asymmetric quantum cloning problem. This concerns the quantum cloning of any quantum state to two approximate clones of different qualities. Measuring the cloning performance with the probabilities of the mixed clone states as the figure of merit, we show that the physical region is a union of ellipses in the plane. The study of these regions has consequences for the eavesdropping on quantum cryptography, and a wide variety of tasks. Equivalently, we characterize the region of quantum-state compatibility of two possibly different isotropic states, considering, for the first time, negative figures of merit. [ABSTRACT FROM AUTHOR]

We describe a generalization of Hashimoto and Kurano's Cauchy filtration for divided powers algebras. This filtration is then used to provide a cellular structure for generalized Schur algebras associated to an arbitrary cellular algebra. Applications to the cellularity of wreath product algebras A ≀ S d are also considered. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a unitary matrix-valued process which is closely related to the Hermitian matrix-Jacobi process. It is precisely defined as the product of a deterministic self-adjoint symmetry and a randomly-rotated one by a unitary Brownian motion. Using stochastic calculus and the action of the symmetric group on tensor powers, we derive an ordinary differential equation for the moments of its fixed-time marginals. Next, we derive an expression of these moments which involves a unitary bridge between our unitary process and another independent unitary Brownian motion. This bridge motivates and allows to write a second direct proof of the obtained moment expression. [ABSTRACT FROM AUTHOR]

Based on the work of Vershik (J Sov Math 59(5):1029–1040, 1992), we introduce two new combinatorial identities. We show how these identities can be used to prove a new hook-content identity. The main motivation for deriving this identity was a particular optimization problem in the field of quantum information processing. [ABSTRACT FROM AUTHOR]

The aim of this note is to compare work of Formanek [7] on a certain construction of central polynomials with that of Collins [3] on integration on unitary groups. These two quite disjoint topics share the construction of the same function on the symmetric group, which the second author calls Weingarten function. By joining these two approaches we succeed in giving a simplified and very natural presentation of both Formanek and Collins's Theory. [ABSTRACT FROM AUTHOR]

We define the affine VW supercategory , which arises from studying the action of the periplectic Lie superalgebra p (n) on the tensor product M ⊗ V ⊗ a of an arbitrary representation M with several copies of the vector representation V of p (n) . It plays a role analogous to that of the degenerate affine Hecke algebras in the context of representations of the general linear group; the main obstacle was the lack of a quadratic Casimir element in p (n) ⊗ p (n) . When M is the trivial representation, the action factors through the Brauer supercategory s B r . Our main result is an explicit basis theorem for the morphism spaces of and, as a consequence, of s B r . The proof utilises the close connection with the representation theory of p (n) . As an application we explicitly describe the centre of all endomorphism algebras, and show that it behaves well under the passage to the associated graded and under deformation. [ABSTRACT FROM AUTHOR]

We present an application of the Schur–Weyl duality in the one-dimensional Hubbard model in the case of a half-filled system of any number of atoms. We replace the actions of the dual symmetric and unitary groups in the whole 4 N -dimensional Hilbert space by the actions of the dual groups in the spin and pseudo-spin spaces. The calculations significantly reduce the dimension of the eigenproblem of the one-dimensional Hubbard model. [ABSTRACT FROM AUTHOR]

In this paper we construct a family of exact functors from the category of Whittaker modules of the simple complex Lie algebra of type A n to the category of finite-dimensional modules of the graded affine Hecke algebra of type A ℓ. Using results of Backelin [2] and of Arakawa-Suzuki [1] , we prove that these functors map standard modules to standard modules (or zero) and simple modules to simple modules (or zero). Moreover, we show that each simple module of the graded affine Hecke algebra appears as the image of a simple Whittaker module. Since the Whittaker category contains the BGG category O as a full subcategory, our results generalize results of Arakawa-Suzuki [1] , which in turn generalize Schur-Weyl duality between finite-dimensional representations of SL n (C) and representations of the symmetric group S n. [ABSTRACT FROM AUTHOR]

We construct a new basis for a slim cyclotomic q -Schur algebra S m (n , r) via symmetric polynomials in Jucys–Murphy operators of the cyclotomic Hecke algebra H m (r). We show that this basis, labelled by matrices, is not the double coset basis when H m (r) is the Hecke algebra of a Coxeter group, but coincides with the double coset basis for the corresponding group algebra, the Hecke algebra at q = 1. As further applications, we then discuss the cyclotomic Schur–Weyl duality at the integral level. This also includes a category equivalence and a classification of simple objects. [ABSTRACT FROM AUTHOR]

We investigate the spin-Brauer diagram algebra, denoted S B n (δ) , that arises from studying an analogous form of Schur–Weyl duality for the action of the pin group on V ⊗ n ⊗ Δ . Here, V is the standard N-dimensional complex representation of Pin (N) and Δ is the spin representation. When δ = N is a positive integer, we define a surjective map S B n (N) ↠ End Pin (N) (V ⊗ n ⊗ Δ) and show it is an isomorphism for N ≥ 2 n . We show S B n (δ) is a cellular algebra and use cellularity to characterize its irreducible representations. [ABSTRACT FROM AUTHOR]

Following the general idea of Schur–Weyl scheme and using two suitable symmetric groups (instead of one), we define so called decomposition tensor of tensor representations for finite and infinite symmetric group. This makes more explicit the classical problem of decomposing tensor representations of these groups into irreducible components. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a notion of affine walled Brauer–Clifford superalgebras B C r , t aff is introduced over an arbitrary integral domain R containing 2 − 1. These superalgebras can be considered as affinization of walled Brauer superalgebras in [9]. By constructing infinitely many homomorphisms from B C r , t aff to a class of level two walled Brauer–Clifford superalgebras over C , we prove that B C r , t aff is free over R with infinite rank. We explain that any finite dimensional irreducible B C r , t aff -module over an algebraically closed field F of characteristic not 2 factors through a cyclotomic quotient of B C r , t aff , called a cyclotomic (or level k) walled Brauer–Clifford superalgebra B C k , r , t. Using a previous method on cyclotomic walled Brauer algebras in [15] , we prove that B C k , r , t is free over R with super rank (k r + t 2 r + t − 1 (r + t) ! , k r + t 2 r + t − 1 (r + t) !) if and only if it is admissible in the sense of Definition 6.4. [ABSTRACT FROM AUTHOR]

The classical case of Schur–Weyl duality states that the actions of the group algebras of GLn and Sd on the dth-tensor power of a free module of finite rank centralize each other. We show that Schur–Weyl duality holds for commutative rings where enough scalars can be chosen whose non-zero differences are invertible. This implies all the known cases of Schur–Weyl duality so far. We also show that Schur–Weyl duality fails for and for any finite field when d is sufficiently large. [ABSTRACT FROM AUTHOR]

Let g be a complex simple Lie algebra and U q L g the corresponding quantum affine algebra. We construct a functor F θ between finite-dimensional modules over a quantum symmetric pair subalgebra of affine type U q k ⊂ U q L g and an orientifold KLR algebra arising from a framed quiver with a contravariant involution, providing a boundary analogue of the Kang-Kashiwara-Kim-Oh generalized Schur-Weyl duality. With respect to their construction, our combinatorial model is further enriched with the poles of a trigonometric K-matrix intertwining the action of U q k on finite-dimensional U q L g -modules. By construction, F θ is naturally compatible with the Kang-Kashiwara-Kim-Oh functor in that, while the latter is a functor of monoidal categories, F θ is a functor of module categories. Relying on a suitable isomorphism à la Brundan-Kleshchev-Rouquier, we prove that F θ recovers the Schur-Weyl dualities due to Fan-Lai-Li-Luo-Wang-Watanabe in quasi-split type AIII. [ABSTRACT FROM AUTHOR]

We establish the Schur-Weyl type duality between double affine Hecke algebras and quantum toroidal superalgebras, generalizing the well known result of Vasserot-Varagnolo [26] to the super case. [ABSTRACT FROM AUTHOR]

We use the fusion formulas of the symmetric group and of the Hecke algebra to construct solutions of the Yang-Baxter equation on irreducible representations of $\mathfrak {gl}_{N}$ , $\mathfrak {gl}_{N|M}$ , $U_{q}(\mathfrak {gl}_{N})$ and $U_{q}(\mathfrak {gl}_{N|M})$ . The solutions are obtained via the fusion procedure for the Yang-Baxter equation, which is reviewed in a general setting. Distinguished invariant subspaces on which the fused solutions act are also studied in the general setting, and expressed, in general, with the help of a fusion function. Only then, the general construction is specialised to the four situations mentioned above. In each of these four cases, we show how the distinguished invariant subspaces are identified as irreducible representations, using the relevant fusion formula combined with the relevant Schur-Weyl duality. [ABSTRACT FROM AUTHOR]

Combinatorial formulas for the moments of the Brownian motion on classical compact Lie groups are obtained. These expressions are deformations of formulas of B. Collins and P. ´Sniady for moments of the Haar measure and yield a proof of the First Fundamental Theorem of invariant theory and of classical Schur-Weyl dualities based on stochastic calculus. [ABSTRACT FROM AUTHOR]

In this paper, we give an explicit construction of a quasi-idempotent in the $q$-rook monoid algebra $R_{n}(q)$ and show that it generates the whole annihilator of the tensor space $U^{\otimes n}$ in $R_{n}(q)$. [ABSTRACT FROM AUTHOR]

We give explicit isomorphisms between simple modules of degenerate cyclotomic Hecke algebras defined via various cellular bases. A special case gives a generalized Mullineux involution in the degenerate case. [ABSTRACT FROM AUTHOR]

We prove an analogue of Schur-Weyl duality for the space of homogeneous Lie polynomials of degree r in n variables [ABSTRACT FROM AUTHOR]

The partition algebra $$\mathsf {P}_k(n)$$ and the symmetric group $$\mathsf {S}_n$$ are in Schur-Weyl duality on the k-fold tensor power $$\mathsf {M}_n^{\otimes k}$$ of the permutation module $$\mathsf {M}_n$$ of $$\mathsf {S}_n$$ , so there is a surjection $$\mathsf {P}_k(n) \rightarrow \mathsf {Z}_k(n) := \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})$$ , which is an isomorphism when $$n \ge 2k$$ . We prove a dimension formula for the irreducible modules of the centralizer algebra $$\mathsf {Z}_k(n)$$ in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities of the irreducible $$\mathsf {S}_n$$ -modules in $$\mathsf {M}_n^{\otimes k}$$ . Our dimension expressions hold for any $$n \ge 1$$ and $$k\ge 0$$ . Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on $$\mathsf {M}_n^{\otimes k}$$ and the quasi-partition algebra corresponding to tensor powers of the reflection representation of $$\mathsf {S}_n$$ . [ABSTRACT FROM AUTHOR]

Let F = { 0 , 1 , 2 , 3 } and define the set K = { K 0 , K 1 , K 2 } of relations on F such that ( x , y ) ∈ K i if and only if x − y ≡ ± i ( mod 4 ) . Let n be a positive integer. We consider the Lee association scheme L ( n ) over Z 4 which is the extension of length n of the initial scheme ( F , K ) . Let T denote the Terwilliger algebra of L ( n ) with respect to the zero codeword of length n . We show that T is generated by a homomorphic image of the universal enveloping algebra of the Lie algebra sl 3 ( C ) and the center Z ( T ) . Furthermore, we determine the irreducible modules for T using the Schur–Weyl duality. [ABSTRACT FROM AUTHOR]

Walks on the representation graph $${\mathcal{R}_{\mathsf{V}}}$$ ( G) determined by a group G and a G-module V are related to the centralizer algebras of the action of G on the tensor powers $${\mathsf{V}^{\otimes k}}$$ via Schur-Weyl duality. This paper explores that connection when the group is $${\mathbb{Z}^{n}_{2}}$$ and the module V is chosen so the representation graph is the n-cube. We describe a basis for the centralizer algebras in terms of labeled partition diagrams. We obtain an expression for the number of walks by counting certain partitions and determine the exponential generating functions for the number of walks. [ABSTRACT FROM AUTHOR]