Weyl Duality Research Articles

Cylindric symmetric functions and positivity

We introduce new families of cylindric symmetric functions as subcoalgebras in the ring of symmetric functions Λ (viewed as a Hopf algebra) which have non-negative structure constants. Combinatorially these cylindric symmetric functions are defined as weighted sums over cylindric reverse plane partitions or - alternatively - in terms of sets of affine permutations. We relate their combinatorial definition to an algebraic construction in terms of the principal Heisenberg subalgebra of the affine Lie algebra slˆn and a specialised cyclotomic Hecke algebra. Using Schur–Weyl duality we show that the new cylindric symmetric functions arise as matrix elements of Lie algebra elements in the subspace of symmetric tensors of a particular level-0 module which can be identified with the small quantum cohomology ring of the k-fold product of projective space. The analogous construction in the subspace of alternating tensors gives the known set of cylindric Schur functions which are related to the small quantum cohomology ring of Grassmannians. We prove that cylindric Schur functions form a subcoalgebra in Λ whose structure constants are the 3-point genus 0 Gromov–Witten invariants. We show that the new families of cylindric functions obtained from the subspace of symmetric tensors also share the structure constants of a symmetric Frobenius algebra, which we define in terms of tensor multiplicities of the generalised symmetric group G(n,1,k).

Slim cyclotomic q-Schur algebras

We construct a new basis for a slim cyclotomic q-Schur algebra Sm(n,r) via symmetric polynomials in Jucys–Murphy operators of the cyclotomic Hecke algebra Hm(r). We show that this basis, labelled by matrices, is not the double coset basis when Hm(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.

Shell Tableaux: A Set Partition Analog of Vacillating Tableaux

Schur–Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analog of Schur–Weyl duality for the group of unipotent upper triangular matrices over a finite field. In this case, the character theory of these upper triangular matrices is "wild" or unattainable. Thus we employ a generalization, known as supercharacter theory, that creates a striking variation on the character theory of the symmetric group with combinatorics built from set partitions. In this paper, we present a combinatorial formula for calculating a restriction and induction of supercharacters based on statistics of set partitions and seashell inspired diagrams. We use these formulas to create a graph that encodes the decomposition of a tensor space, and develop an analog of Young tableaux, known as shell tableaux, to index paths in this graph.

The Rectangular Representation of the Double Affine Hecke Algebra via Elliptic Schur–Weyl Duality

Abstract Given a module $M$ for the algebra ${\mathcal{D}}_{\mathtt{q}}(G)$ of quantum differential operators on $G$, and a positive integer $n$, we may equip the space $F_n^G(M)$ of invariant tensors in $V^{\otimes n}\otimes M$, with an action of the double affine Hecke algebra of type $A_{n-1}$. Here $G= SL_N$ or $GL_N$, and $V$ is the $N$-dimensional defining representation of $G$. In this paper, we take $M$ to be the basic ${\mathcal{D}}_{\mathtt{q}}(G)$-module, that is, the quantized coordinate algebra $M= {\mathcal{O}}_{\mathtt{q}}(G)$. We describe a weight basis for $F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$ combinatorially in terms of walks in the type $A$ weight lattice, and standard periodic tableaux, and subsequently identify $F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$ with the irreducible “rectangular representation” of height $N$ of the double affine Hecke algebra.

Presentations of categories of modules using the Cautis–Kamnitzer–Morrison principle

We study commuting actions of a complex reductive Lie algebra, \mathfrak g , and an algebra A on a finite dimensional vector space V . We outline how such commuting actions can be used to give a presentation of the category of A -modules isomorphic to \mathfrak g -weight spaces of V . This generalizes a result of Cautis–Kamnitzer–Morrison (2014) in the non-quantum setting. As examples, we apply this to classical Schur–Weyl duality and the Brauer Schur–Weyl dualities. In particular we obtain a diagrammatic presentation, in terms of generators and relations, of the category of permutation modules of the symmetric group, S_d . This is used to give a new description of the Kronecker product of the S_d -equivariant morphisms between permutation modules.

Affine Brauer category and parabolic category $${\mathcal {O}}$$ O in types B, C, D

A strict monoidal category referred to as affine Brauer category $$\mathcal {AB}$$ is introduced over a commutative ring $$\kappa $$ containing multiplicative identity 1 and invertible element 2. We prove that morphism spaces in $$\mathcal {AB}$$ are free over $$\kappa $$ . The cyclotomic (or level k) Brauer category $$\mathcal {CB}^f(\omega )$$ is a quotient category of $$\mathcal {AB}$$ . We prove that any morphism space in $$\mathcal {CB}^f(\omega )$$ is free over $$\kappa $$ with maximal rank if and only if the $${\mathbf {u}}$$ -admissible condition holds in the sense of (1.32). Affine Nazarov–Wenzl algebras (Nazarov in J Algebra 182(3):664–693, 1996) and cyclotomic Nazarov–Wenzl algebras (Ariki et al. in Nagoya Math J 182:47–134, 2006) will be realized as certain endomorphism algebras in $$\mathcal {AB}$$ and $$\mathcal {CB}^f(\omega )$$ , respectively. We will establish higher Schur–Weyl duality between cyclotomic Nazarov–Wenzl algebras and parabolic BGG categories $${\mathcal {O}}$$ associated to symplectic and orthogonal Lie algebras over the complex field $$\mathbb C$$ . This enables us to use standard arguments in (Anderson et al. in Pac J Math 292(1):21–59, 2018; Rui and Song in Math Zeit 280(3–4):669–689, 2015; Rui and Song in J Algebra 444:246–271, 2015), to compute decomposition matrices of cyclotomic Nazarov–Wenzl algebras. The level two case was considered by Ehrig and Stroppel in (Adv. Math. 331:58–142, 2018).

Schur–Weyl duality over commutative rings

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.

Auslander–Reiten quiver and representation theories related to KLR-type Schur–Weyl duality

We introduce new partial orders on the sequence positive roots and study the statistics of the poset by using Auslander–Reiten quivers for finite type ADE. Then we can prove that the statistics provide interesting information on the representation theories of KLR-algebras, quantum groups and quantum affine algebras including Dorey’s rule, bases theory for quantum groups, and denominator formulas between fundamental representations. As applications, we prove Dorey’s rule for quantum affine algebras $$U_q(E_{6,7,8}^{(1)})$$ and partial information of denominator formulas for $$U_q(E_{6,7,8}^{(1)})$$ . We also suggest conjecture on complete denominator formulas for $$U_q(E_{6,7,8}^{(1)})$$ .

The Integral Quantum Loop Algebra of $\mathfrak {gl}_{n}$

Abstract We will construct the Lusztig form for the quantum loop algebra of $\mathfrak {gl}_{n}$ by proving the conjecture [4, 3.8.6] and establish partially the Schur–Weyl duality at the integral level in this case. We will also investigate the integral form of the modified quantum affine $\mathfrak {gl}_{n}$ by introducing an affine stabilisation property and will lift the canonical bases from affine quantum Schur algebras to a canonical basis for this integral form. As an application of our theory, we will also discuss the integral form of the modified extended quantum affine $\mathfrak {sl}_{n}$ and construct its canonical basis to provide an alternative algebra structure related to a conjecture of Lusztig in [29, §9.3], which has been already proved in [34].

Annular Khovanov homology and knotted Schur–Weyl representations

Let $\mathbb{L}\subset A\times I$ be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of $\mathfrak{sl}_{2}(\wedge )$, the exterior current algebra of $\mathfrak{sl}_{2}$. When $\mathbb{L}$ is an $m$-framed $n$-cable of a knot $K\subset S^{3}$, its sutured annular Khovanov homology carries a commuting action of the symmetric group $\mathfrak{S}_{n}$. One therefore obtains a ‘knotted’ Schur–Weyl representation that agrees with classical $\mathfrak{sl}_{2}$ Schur–Weyl duality when $K$ is the Seifert-framed unknot.

Integration formulas for Brownian motion on classical compact Lie groups

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. Śniady 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.

Entanglement classification in the noninteracting Fermi gas

In this paper, entanglement classification shared among the spins of localized fermions in the noninteracting Fermi gas is studied. It is proven that the Fermi gas density matrix is block diagonal on the basis of the projection operators to the irreducible representations of symmetric group [Formula: see text]. Every block of density matrix is in the form of the direct product of a matrix and identity matrix. Then it is useful to study entanglement in every block of density matrix separately. The basis of corresponding Hilbert space are identified from the Schur–Weyl duality theorem. Also, it can be shown that the symmetric part of the density matrix is fully separable. Then it has been shown that the entanglement measure which is introduced in Eltschka et al. [New J. Phys. 10, 043104 (2008)] and Guhne et al. [New J. Phys. 7, 229 (2005)], is zero for the even [Formula: see text] qubit Fermi gas density matrix. Then by focusing on three spin reduced density matrix, the entanglement classes have been investigated. In three qubit states there is an entanglement measure which is called 3-tangle. It can be shown that 3-tangle is zero for three qubit density matrix, but the density matrix is not biseparable for all possible values of its parameters and its eigenvectors are in the form of W-states. Then an entanglement witness for detecting non-separable state and an entanglement witness for detecting nonbiseparable states, have been introduced for three qubit density matrix by using convex optimization problem. Finally, the four spin reduced density matrix has been investigated by restricting the density matrix to the irreducible representations of [Formula: see text]. The restricted density matrix to the subspaces of the irreducible representations: [Formula: see text], [Formula: see text] and [Formula: see text] are denoted by [Formula: see text], [Formula: see text] and [Formula: see text], respectively. It has been shown that some highly entangled classes (by using the results of Miyake [Phys. Rev. A 67, 012108 (2003)] for entanglement classification) do not exist in the blocks of density matrix [Formula: see text] and [Formula: see text], so these classes do not exist in the total Fermi gas density matrix.

Turner doubles and generalized Schur algebras

Turner's Conjecture describes all blocks of symmetric groups and Hecke algebras up to derived equivalence in terms of certain double algebras. With a view towards a proof of this conjecture, we develop a general theory of Turner doubles. In particular, we describe doubles as explicit maximal symmetric subalgebras of certain generalized Schur algebras and establish a Schur–Weyl duality with wreath product algebras.

Vust's Theorem and higher level Schur–Weyl duality for types B, C and D

Let G be a complex linear algebraic group, g=Lie(G) its Lie algebra and e∈g a nilpotent element. Vust's Theorem says that in case of G=GL(V), the algebra EndGe(V⊗d), where Ge⊂G is the stabilizer of e under the adjoint action, is generated by the image of the natural action of d-th symmetric group Sd and the linear maps {1⊗(i−1)⊗e⊗1⊗(d−i)|i=1,…,d}. In this paper, we give an analogue of Vust's Theorem for G=O(V) and SP(V) when the nilpotent elements e satisfy that G⋅e‾ is normal. As an application, we study the higher Schur–Weyl duality in the sense of [4] for types B, C and D, which establishes a relationship between W-algebras and degenerate affine braid algebras.

Fusion Formulas and Fusion Procedure for the Yang-Baxter Equation

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.

Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

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 \(\mathsf {S}_n\).

Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras IV

Let $U'_q(\mathfrak{g})$ be a twisted affine quantum group of type $A_{N}^{(2)}$ or $D_{N}^{(2)}$ and let $\mathfrak{g}_{0}$ be the finite-dimensional simple Lie algebra of type $A_{N}$ or $D_{N}$. For a Dynkin quiver of type $\mathfrak{g}_{0}$, we define a full subcategory ${\mathcal C}_{Q}^{(2)}$ of the category of finite-dimensional integrable $U'_q(\mathfrak{g})$-modules, a twisted version of the category ${\mathcal C}_{Q}$ introduced by Hernandez and Leclerc. Applying the general scheme of affine Schur-Weyl duality, we construct an exact faithful KLR-type duality functor ${\mathcal F}_{Q}^{(2)}: Rep(R) \rightarrow {\mathcal C}_{Q}^{(2)}$, where $Rep(R)$ is the category of finite-dimensional modules over the quiver Hecke algebra $R$ of type $\mathfrak{g}_{0}$ with nilpotent actions of the generators $x_k$. We show that ${\mathcal F}_{Q}^{(2)}$ sends any simple object to a simple object and induces a ring isomorphism $K(Rep(R)) \simeq K({\mathcal C}_{Q}^{(2)})$.

On Lee association schemes over [formula omitted] and their Terwilliger algebra

Let F={0,1,2,3} and define the set K={K0,K1,K2} of relations on F such that (x,y)∈Ki if and only if x−y≡±i(mod 4). Let n be a positive integer. We consider the Lee association scheme L(n) over Z4 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 sl3(C) and the center Z(T). Furthermore, we determine the irreducible modules for T using the Schur–Weyl duality.

Auslander–Reiten quiver of type D and generalized quantum affine Schur–Weyl duality

We first provide an explicit combinatorial description of the Auslander–Reiten quiver ΓQ of finite type D. Then we can investigate the categories of finite dimensional representations over the quantum affine algebra Uq′(Dn+1(i))(i=1,2) and the quiver Hecke algebra RDn+1 associated to Dn+1(n≥3), by using the combinatorial description and the generalized quantum affine Schur–Weyl duality functor. As applications, we can prove that Dorey's rule holds for the category Rep(RDn+1) and prove an interesting difference between multiplicity free positive roots and multiplicity non-free positive roots.

Schur–Weyl duality for the Brauer algebra and the ortho-symplectic Lie superalgebra

We give a proof of a Schur–Weyl duality statement between the Brauer algebra and the ortho-symplectic Lie superalgebra \(\mathfrak {osp}(V)\).