Central Algebras Research Articles

The Terwilliger algebras of Odd graphs and Doubled Odd graphs

The Terwilliger algebras of Odd graphs and Doubled Odd graphs

Solutions to the constant Yang–Baxter equation: additive charge conservation in three dimensions

We find all solutions to the constant Yang–Baxter equation R 12 R 13 R 23 = R 23 R 13 R 12 in three dimensions, subject to an additive charge-conservation (ACC) ansatz. This ansatz is a generalization of (strict) charge-conservation, for which a complete classification in all dimensions was recently obtained. ACC introduces additional sector-coupling parameters—in three dimensions, there are four such parameters. In the generic dimension 3 case, in which all of the four parameters are non-zero, we find there is a single three parameter family of solutions. We give a complete analysis of this solution, giving the structure of the centralizer (symmetry) algebra in all orders. We also solve the remaining cases with three, two or one non-zero sector-coupling parameter(s).

The Terwilliger algebras of the group association schemes of three metacyclic groups

AbstractFor any finite group , the Terwilliger algebra of the group association scheme satisfies the following inclusions: , where is a specific vector space and is the centralizer algebra of the permutation representation of induced by the action of conjugation. The group is said to be triply transitive if . In this paper, we determine the dimensions of and for being , and , and show that and are triply transitive. Additionally, we give a complete characterization of the Wedderburn components of the Terwilliger algebras of , and when they are triply transitive.

The partial Temperley–Lieb algebra and its representations

In this paper, we give a combinatorial description of a new diagram algebra, the partial Temperley–Lieb algebra, arising as the generic centralizer algebra \mathrm{End}_{\mathbf{U}_q(\mathfrak{gl}_2)}(V^{\otimes k}) , where {V = V(0) \oplus V(1)} is the direct sum of the trivial and natural module for the quantized enveloping algebra \mathbf{U}_q(\mathfrak{gl}_2) . It is a proper subalgebra of the Motzkin algebra (the \mathbf{U}_q(\mathfrak{sl}_2) -centralizer) of Benkart and Halverson. We prove a version of Schur–Weyl duality for the new algebras, and describe their generic representation theory.

Terwilliger algebras and some related algebras defined by finite connected simple graphs

Terwilliger algebras and some related algebras defined by finite connected simple graphs

Matrix and tensor witnesses of hidden symmetry algebras

Permutation group algebras, and their generalizations called permutation centralizer algebras (PCAs), play a central role as hidden symmetries in the combinatorics of large N gauge theories and matrix models with manifest continuous gauge symmetries. Polynomial functions invariant under the manifest symmetries are the observables of interest and have applications in AdS/CFT. We compute such correlators in the presence of matrix or tensor witnesses, which by definition, can include a matrix or tensor field appearing as a coupling in the action (i.e a spurion) or as a classical (un-integrated) field in the observables, appearing alongside quantum (integrated) fields. In both matrix and tensor cases we find that two-point correlators of general gauge-invariant observables can be written in terms of gauge invariant functions of the witness fields, with coefficients given by structure constants of the associated PCAs. Fourier transformation on the relevant PCAs, relates combinatorial bases to representation theoretic bases. The representation theory basis elements obey orthogonality results for the two-point correlators which generalise known orthogonality relations to the case with witness fields. The new orthogonality equations involve two representation basis elements for observables as input and a representation basis observable constructed purely from witness fields as the output. These equations extend known equations in the super-integrability programme initiated by Mironov and Morozov, and are a direct physical realization of the Wedderburn-Artin decompositions of the hidden permutation centralizer algebras of matrix/tensor models.

Multiplication formulas and isomorphism theorem of ıSchur superalgebras

We introduce the notion of ı 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 ı Schur superalgebra are obtained to construct their canonical bases. Furthermore, we established an isomorphism theorem between the ı Scuhr superalgebras and the q -Schur superalgebras of type A, which helps us derive semisimplicity criteria of the ı Schur superalgebras.

An $$\mathfrak {sl}_2$$-type tensor category for the Virasoro algebra at central charge 25 and applications

Let $\mathcal{O}_{25}$ be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge $25$ whose composition factors are the irreducible quotients of reducible Verma modules. We show that $\mathcal{O}_{25}$ is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian $3$-cocycle twist of the category of finite-dimensional $\mathfrak{sl}_2$-modules. We also show that this $\mathfrak{sl}_2$-type subcategory is braid-reversed tensor equivalent to a similar category for the Virasoro algebra at central charge $1$. As an application, we construct a simple conformal vertex algebra which contains the Virasoro vertex operator algebra of central charge $25$ as a $PSL_2(\mathbb{C})$-orbifold. We also use our results to study Arakawa's chiral universal centralizer algebra of $SL_2$ at level $-1$, showing that it has a symmetric tensor category of representations equivalent to $\mathrm{Rep}\,PSL_2(\mathbb{C})$. This algebra is an extension of the tensor product of Virasoro vertex operator algebras of central charges $1$ and $25$, analogous to the modified regular representations of the Virasoro algebra constructed earlier for generic central charges by I. Frenkel-Styrkas and I. Frenkel-M. Zhu.

Doubly Stochastic Matrices and Schur-Weyl Duality for Partition Algebras

We prove that the permutations of $\{1,\dots, n\}$ having an increasing (resp., decreasing) subsequence of length $n-r$ index a subset of the set of all $r$th Kronecker powers of $n \times n$ permutation matrices which is a basis for the linear span of that set. Thanks to a known Schur-Weyl duality, this gives a new basis for the centralizer algebra of the partition algebra acting on the $r$th tensor power of a vector space. We give some related results on the set of doubly stochastic matrices in that algebra.

Centralizer matrix algebras and symmetric polynomials of partitions

For a field R of characteristic p ≥ 0 and a matrix c in the full n × n matrix algebra M n ( R ) over R , let S n ( c , R ) be the centralizer algebra of c in M n ( R ) . We show that S n ( c , R ) is a Frobenius-finite, 1-Auslander-Gorenstein, and gendo-symmetric algebra, and that the extension S n ( c , R ) ⊆ M n ( R ) is separable and Frobenius . Further, we study the isomorphism problem of invariant matrix algebras . Let σ be a permutation in the symmetric group Σ n and c σ the corresponding permutation matrix in M n ( R ) . We give sufficient and necessary conditions for the invariant algebra S n ( c σ , R ) to be semisimple . If R is an algebraically closed field , we establish a combinatoric characterization of when two semisimple invariant R -algebras are isomorphic in terms of the cycle types of permutations.

Howe duality of the symmetric group and a multiset partition algebra

We introduce the multiset partition algebra, that has basis elements indexed by multiset partitions, where x is an indeterminate and r and k are non-negative integers. This algebra can be realized as a diagram algebra that generalizes the partition algebra. When x = n is an integer greater or equal to 2r, we show that is isomorphic to a centralizer algebra of the symmetric group, Sn , acting on the polynomial ring in the variables For the centralizer algebra we give a formula for the dimensions of the irreducible representations and a formula for the branching rule. We also show that the Kronecker coefficients occur as multiplicities in the restrictions of these irreducible modules.

Traces on diagram algebras II: centralizer algebras of easy groups and new variations of the Young graph

In continuation of our recent work arXiv:2006.07312, we classify the extremal traces on infinite diagram algebras that appear in the context of Schur-Weyl duality for Banica and Speicher's easy groups. We show that the branching graphs of these algebras describe walks on new variations of the Young graph which describe curious ways of growing Young diagrams. As a consequence, we prove that the extremal traces on generic rook-Brauer algebras are always extensions of extremal traces on the group algebra $\mathbb{C}[S_{\infty}]$ of the infinite symmetric group. Moreover, we conjecture that the same is true for generic parameter deformations of the centralizers of the hyperoctahedral group and we reduce this conjecture to a conceptually much simpler numerical statement. Lastly, we address the trace classification problem for the Schur-Weyl dual of the halfliberated orthogonal group $O_N^*$, in which case extremal traces are always extensions of extremal traces on $\mathbb{C}[S_{\infty} \times S_{\infty}]$. Our approach relies on methods developed by Vershik and Nikitin.

Tensor representations for the Drinfeld double of the Taft algebra

Over an algebraically closed field $\mathbb k$ of characteristic zero, the Drinfeld double $D_n$ of the Taft algebra that is defined using a primitive $n$th root of unity $q \in \mathbb k$ for $n \geq 2$ is a quasitriangular Hopf algebra. Kauffman and Radford have shown that $D_n$ has a ribbon element if and only if $n$ is odd, and the ribbon element is unique; however there has been no explicit description of this element. In this work, we determine the ribbon element of $D_n$ explicitly. For any $n \geq 2$, we use the R-matrix of $D_n$ to construct an action of the Temperley-Lieb algebra $\mathsf{TL}_k(\xi)$ with $\xi = -(q^{\frac{1}{2}}+q^{-\frac{1}{2}})$ on the $k$-fold tensor power $V^{\otimes k}$ of any two-dimensional simple $D_n$-module $V$. This action is known to be faithful for arbitrary $k \geq 1$. We show that $\mathsf{TL}_k(\xi)$ is isomorphic to the centralizer algebra $\text{End}_{D_n}(V^{\otimes k})$ for $1 \le k \le 2n-2$.

Note on the permutation group associated to E-polynomials

This is a continuation of our project which focuses on E-polynomials and the related combinatorics. A pair of groups appearing in the definition of E-polynomials yields the permutation group. In this paper, we determine the multi-matrix structures of the centralizer algebras of the tensor representations of this permutation group.

The Terwilliger algebra of the halved [formula omitted]-cube from the viewpoint of its automorphism group action

Let 1 2 H ( n , 2 ) denote the halved n -cube with vertex set X and let T ≔ T ( x 0 ) denote the Terwilliger algebra of 1 2 H ( n , 2 ) with respect to a fixed vertex x 0 ∈ X . In this paper, we assume n ≥ 6 . We first characterize T by considering the action of the automorphism group of 1 2 H ( n , 2 ) on the set X × X × X . We show that T coincides with the centralizer algebra of the stabilizer of x 0 in the automorphism group, and display three subalgebras of T further. Then we study the homogeneous components of V ≔ ℂ X , each of which is a nonzero subspace of V spanned by the irreducible T -modules that are isomorphic. We give a computable basis for any homogeneous component of V . Finally, we describe the decomposition of T via its block-diagonalization and give a basis for the center of T by using the above homogeneous components of V .

Triangular matrix algebras over affine quasi-hereditary algebras

We prove that the triangular matrix algebra Λ = ( H 0 H H ) is an affine quasi-hereditary algebra if and only if H is an affine quasi-hereditary algebra. Moreover, the category of Δ-good Λ-modules, the global dimension and the characteristic tilting module of Λ are described by using the corresponding ones of H . In the appendix, we prove that certain centralizer algebra and quotient algebra of an affine quasi-hereditary algebra are affine quasi-hereditary.

BRAIDED COMMUTATIVE ALGEBRAS OVER QUANTIZED ENVELOPING ALGEBRAS

We produce braided commutative algebras in braided monoidal categories by generalizing Davydov’s full center construction of commutative algebras in centers of monoidal categories. Namely, we build braided commutative algebras in relative monoidal centers {mathcal{Z}}_{mathrm{mathcal{B}}}left(mathcal{C}right) from algebras in ℬ-central monoidal categories mathcal{C} , where ℬ is an arbitrary braided monoidal category; Davydov’s (and previous works of others) take place in the special case when ℬ is the category of vector spaces {mathbf{Vect}}_{mathbbm{K}} over a field mathbbm{K} . Since key examples of relative monoidal centers are suitable representation categories of quantized enveloping algebras, we supply braided commutative module algebras over such quantum groups. One application of our work is that we produce Morita invariants for algebras in ℬ-central monoidal categories. Moreover, for a large class of ℬ-central monoidal categories, our braided commutative algebras arise as a braided version of centralizer algebras. This generalizes the fact that centers of algebras in {mathbf{Vect}}_{mathbbm{K}} serve as Morita invariants. Many examples are provided throughout.

McKay Centralizer Algebras

For a finite subgroup G of the special unitary group SU2, we study the centralizer algebra Zk(G) = EndG(V⊗k) of G acting on the k-fold tensor product of its defining representation V = C2. The McKay corre- spondence relates the representation theory of these groups to an associated affine Dynkin diagram, and we use this connection to study the structure and representation theory of Zk(G) via the combinatorics of the Dynkin diagram. When G equals the binary tetrahedral, octahedral, or icosahedral group, we exhibit remarkable connections between Zk (G) and the Martin-Jones set partition algebras.

An insertion algorithm on multiset partitions with applications to diagram algebras

We generalize the Robinson-Schensted-Knuth algorithm to the insertion of two row arrays of multisets. This generalization leads to new enumerative results that have representation theoretic interpretations as decompositions of centralizer algebras and the spaces they act on. In addition, restrictions on the multisets lead to further identities and representation theory analogues. For instance, we obtain a bijection between words of length $k$ with entries in $[n]$ and pairs of tableaux of the same shape with one being a standard Young tableau of size $n$ and the other being a standard multiset tableau of content $[k]$. We also obtain an algorithm from partition diagrams to pairs of a standard tableau and a standard multiset tableau of the same shape, which has the remarkable property that it is well-behaved with respect to restricting a representation to a subalgebra. This insertion algorithm matches recent representation-theoretic results of Halverson and Jacobson.

Non-extremal weight modules for quantized universal enveloping algebras

Abstract For quantized universal enveloping algebras we construct weight modules by inducing representations of the centralizer of the Cartan subalgebra in the quantized universal enveloping algebra. The induced modules arising from finite-dimensional weight modules the centralizer algebra are studied. In particular, we study the induction of one-dimensional modules, and this is related to the study of commutative subalgebras of the centralizer algebra. For the special case of U q ( sl ( 2 , ℂ ) ) we show that we get the admissible unitary representations corresponding to the non-compact real form U q ( su ( 1 , 1 ) ) .