Cohomology of Tanabe algebras

On partition algebras for complex reflection groups

This paper provides a systematization of some recent results in homology of algebras. Our main theorem gives criteria under which the homology of a diagram algebra is isomorphic to the homology of the subalgebra on diagrams having the maximum number of left-to-right connections. From this theorem, we deduce the ‘invertible-parameter’ cases of the Temperley–Lieb and Brauer results of Boyd–Hepworth and Boyd–Hepworth–Patzt. We are also able to give a new proof of Sroka’s theorem that the homology of an odd-strand Temperley–Lieb algebra vanishes, as well as an analogous result for Brauer algebras and an interpretation of both results in the even-strand case. Our proofs are relatively elementary: in particular, no auxiliary chain complexes or spectral sequences are required. We briefly discuss the relationship to cellular algebras in the sense of Graham–Lehrer.

We classify finite order symmetries g of the 14 exceptional unimodal function singularities f in 3 variables, which satisfy a so-called splitting condition. This means that the rank 2 positive subspace in the vanishing homology of f should not be contained in one eigenspace of g?. We also obtain a description of the hyperbolic complex reflection groups appearing as equivariant monodromy groups acting on the hyperbolic eigensubspaces arising. One of the most famous classical results in singularity theory is the Arnold and Brieskorn discovery of the close relationship between simple function singularities and Weyl groups Aμ, Dμ, Eμ [1, 6]. A few years after it, Arnold extended the relationship to simple singularities with the Z2 reflection symmetry and Weyl groups Bμ, Cμ, F4 [2] (see also Slodowy’s book [24]). Consideration of Zm symmetries of simple functions led in [10, 11, 12, 25] to the appearance of Shephard-Todd groups within function singularities. The emphasis there was on realisations of the complex reflection groups as equivariant monodromy groups acting on the appropriate character subspaces in the homology of invariant Milnor fibres, and on the diffeomorphisms between the discriminants of the reflection groups and of the Zm-equivariant functions. A further series of papers [13, 14, 15], on cyclic symmetries of the parabolic functions, brought in similar singularity realisations of certain complex crystallographic groups [22]. In this paper, we are naturally expanding the programme to cyclic symmetries of the 14 exceptional unimodal function singularities on one hand, and complex hyperbolic reflection groups on the other. The basic idea is as follows. In the 3-variable case, the intersection form on the vanishing homology of an exceptional unimodal function f is non-degenerate and has positive signature 2. Assume g is an automorphism of C of finite order m, and our function is g-invariant. Then g acts on the second homology of the Milnor fibre f−1(e), and decomposes it into a direct sum of the character subspaces Hχ, χ = 1, on which g acts as multiplication by χ. Assume the rank 2 positive subspace of the intersection form splits between two character summands. Then the monodromy within a g-invariant versal deformation of f acts as a complex hyperbolic reflection group on each of them. Developing further the technique introduced in papers on cyclically symmetric functions [10, 11, 12], we construct vanishing bases in the hyperbolic summands and obtain the generating reflections as the corresponding Picard-Lefschetz operators. The main result of the paper is a complete classification of the invariant symmetries of the 14 singularities, which split the positive subspace in the vanishing homology, and the description – via constructing the corresponding Dynkin diagrams – of the complex hyperbolic groups arising. All the rank 2 reflection groups obtained projectivise to the triangle groups of the Poincare disk. The task of identification of higher dimensional groups is left for a future paper, along with the consideration of the equivariant symmetry setting. It should be noted that it is the first time when complex hyperbolic reflection groups are appearing in a singularity theory context. The approach introduced may be useful for constructing new complex hyperbolic lattices (cf. [8, 20]). The paper is organised as follows. Section 1 introduces the notion of singularities with symmetry, recalls the definitions and constructions given in [10, 11, 12]. Section 2 contains classification of splitting invariant symmetries of the 14 singularities. In Section 3.3 we construct Dynkin diagrams of the hyperbolic monodromy groups associated with the symmetric functions. Projectivisations of the rank 2 monodromy groups are considered in Section 4. More details of the constructions may be found in [16].

Cellularity of twisted semigroup algebras

The partition algebras P k (x) have been defined in Martin [Martin, P. (1990). Representations of graph temperley Lieb algebras. Pupl. Res. Inst. Math. Sci. 26:485–503] and Jones [Jones, V. F. R. (1993). The Potts model and the symmetric group. In: Subfactors: Proceedings of the Taniguchi Symposium on Operator Algebra, Kyuzeso. River edge, NJ: World Scientific, pp. 259–267, 1994]. We introduce a new class of algebras for every group G called “G-Vertex Colored Partition Algebras,” denoted by P k (x, G), which contain partition algebras P k (x), as subalgebras. We generalized Jones result by showing that for a finite group G, the algebra P k (n, G) is the centralizer algebra of an action of the direct product S n × G on tensor powers of its permutation module. Further we show that these algebras P k (x, G) contain as subalgebras the “G-Colored Partition Algebras P k (x;G)” introduced in Bloss [Bloss, M. (2003). G-colored partition algebras as centralizer algebras of wreath products. J. Algebra 265:690–710].

A concrete description of Hochschild cohomology is the first step toward exploring associative deformations of algebras. In this dissertation, deformation theory, geometry, combinatorics, invariant theory, representation theory, and homological algebra merge in an investigation of Hochschild cohomology of skew group algebras arising from complex reflection groups. Given a linear action of a finite group on a finite dimensional vector space, the skew group algebra under consideration is the semi-direct product of the group with a polynomial ring on the vector space. Each representation of a group defines a different skew group algebra, which may have its own interesting deformations. In this work, we explicitly describe all graded Hecke algebras arising as deformations of the skew group algebra of any finite group acting by the regular representation. We then focus on rank two exceptional complex reflection groups acting by any irreducible representation. We consider in-depth the reflection representation and a nonfaithful rotation representation. Alongside our study of cohomology for the rotation representation, we develop techniques valid for arbitrary finite groups acting by a representation with a central kernel. Additionally, we consider combinatorial questions about reflection length and codimension orderings on complex reflection groups. We give algorithms using character theory to compute reflection length, atoms, and poset relations. Using a mixture of theory, explicit examples, and calculations using the software GAP, we show that Coxeter groups and the infinite family G(m,1,n) are the only irreducible complex reflection groups for which the reflection length and codimension orders coincide. We describe the atoms in the codimension order for the groups G(m,p,n). For arbitrary finite groups, we show that the codimension atoms are contained in the support of every generating set for cohomology, thus yielding information about the degrees of generators for cohomology.

The symmetric group \(\mathsf {S}_n\) and the partition algebra \(\mathsf {P}_k(n)\) centralize one another in their actions on the k-fold tensor power \(\mathsf {M}_n^{\otimes k}\) of the n-dimensional permutation module \(\mathsf {M}_n\) of \(\mathsf {S}_n\). The duality afforded by the commuting actions determines an algebra homomorphism \(\varPhi _{k,n}: \mathsf {P}_k(n) \rightarrow \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\) from the partition algebra to the centralizer algebra \( \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), which is a surjection for all \(k, n \in \mathbb {Z}_{\ge 1}\), and an isomorphism when \(n \ge 2k\). We present results that can be derived from the duality between \(\mathsf {S}_n\) and \(\mathsf {P}_k(n)\), for example, (i) expressions for the multiplicities of the irreducible \(\mathsf {S}_n\)-summands of \(\mathsf {M}_n^{\otimes k}\), (ii) formulas for the dimensions of the irreducible modules for the centralizer algebra \( \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), (iii) a bijection between vacillating tableaux and set-partition tableaux, (iv) identities relating Stirling numbers of the second kind and the number of fixed points of permutations, and (v) character values for the partition algebra \(\mathsf {P}_k(n)\). When \(2k >n\), the map \(\varPhi _{k,n}\) has a nontrivial kernel which is generated as a two-sided ideal by a single idempotent. We describe the kernel and image of \(\varPhi _{k,n}\) in terms of the orbit basis of \(\mathsf {P}_k(n)\) and explain how the surjection \(\varPhi _{k,n}\) can also be used to obtain the fundamental theorems of invariant theory for the symmetric group.

Let V be a free module of rank n over a commutative ring 𝕜. We prove that tensor space V ⊗r satisfies Schur–Weyl duality, regarded as a bimodule for the action of the group algebra of the Weyl group of GL(V) and the partition algebra 𝒫 r (n) over 𝕜. We also prove a similar result for the half partition algebra.

Several finite complex reflection groups have a braid group that is isomorphic to a torus knot group. The reflection group is obtained from the torus knot group by declaring meridians to have order k for some $k\geq 2$ , and meridians are mapped to reflections. We study all possible quotients of torus knot groups obtained by requiring meridians to have finite order. Using the theory of J-groups of Achar and Aubert [‘On rank 2 complex reflection groups’, Comm. Algebra36(6) (2008), 2092–2132], we show that these groups behave like (in general, infinite) complex reflection groups of rank two. The large family of ‘toric reflection groups’ that we obtain includes, among others, all finite complex reflection groups of rank two with a single conjugacy class of reflecting hyperplanes, as well as Coxeter’s truncations of the $3$ -strand braid group. We classify these toric reflection groups and explain why the corresponding torus knot group can be naturally considered as its braid group. In particular, this yields a new infinite family of reflection-like groups admitting braid groups that are Garside groups. Moreover, we show that a toric reflection group has cyclic center by showing that the quotient by the center is isomorphic to the alternating subgroup of a Coxeter group of rank three. To this end we use the fact that the center of the alternating subgroup of an irreducible, infinite Coxeter group of rank at least three is trivial. Several ingredients of the proofs are purely Coxeter-theoretic, and might be of independent interest.

The partition algebra P k (x) has been defined by Martin [Martin, P. (1990). Representations of graph Temperley Lieb algebras. Pupl. Res. Inst. Math. Sci. 26:485–503] and Jones [Jones, V. F. R. (1993). The Potts model and the symmetric group. In: Subfactors, Proceedings of the Taniguchi Symposium on Operator Algebra, Kyuzeso, 1993. River edge, NJ: World Scientific, pp. 259–267], as one having a basis consisting of partitions of 2k objects. We introduce a new class of algebras called “Signed Partition Algebras,” denoted by , which contain partition algebras as subalgebras. The signed partition algebras are a generalization of the partition algebras and signed Brauer algebras [Brauer, R. (1937). On algebras which are connected with the semisimple continuous groups. Ann. Math. 38:854–872; Parvathi, M., Kamaraj, M. (1998). Signed Brauer's algebras. Comm. Algebra 26(3):839–855]. We also give a description in terms of generators and relations. Further we show that, when G = Z 2 the algebra , “The Algebra of G-relations,” introduced by Kodiyalam et al. [Kodiyalam, V., Srinivasan, R., Sunder, V. S. (2000). The algebra of G-relations. Proc. Indian Acad. Sci. (Math. Sci.) 110(3):263–292], may be realized as the centralizer of certain subgroup of the symmetric group action on V ⊗k , and that contains the signed partition algebra , as a subalgebra.

The partition algebra P(q) is a generalization both of the Brauer algebra and the Temperley–Lieb algebra for q-state n-site Potts models, underpining their transfer matrix formulation on the arbitrary transverse lattices. We prove that for arbitrary field k and any element q∈ k the partition algebra P(q) is always cellular in the sense of Graham and Lehrer. Thus the representation theory of P(q) can be determined by applying the developed general representation theory on cellular algebras and symmetric groups. Our result also provides an explicit structure of P(q) for arbitrary field and implies the well-known fact that the Brauer algebra D(q) and the Temperley–Lieb algebra TL(q) are cellular.

Eigenvectors of elements of real and complex reflection groups have been studied byCoxeter, Kostant, Springer, Lehrer and Bessis amongst others, due to their rich geometry, and the extensive information that they yield about the structure of the reflection group and related objects in Lie theory and braid theory. Recently, Kamgarpour proved that if G is an irreducible finite real reflection group of rank n, x is an eigenvector of any element of G with eigenvalue a primitive dth root of unity, and Φ and Φx denote the root systems of G and StabG(x) respectively, then |Φ| − |Φx| ≥ dn, with equality if and only if d is the Coxeter number.In this thesis, we prove the following generalisation of Kamgarpour’s inequality. If G is an irreducible complex reflection group of rank n and x is any eigenvector of an element of G with eigenvalue a primitive dth root of unity, then we havel(π ) − l(π x) ≥ dn,with equality if and only if G is well-generated and d = h. Here, π is the generator of the centre of the pure braid group of G, π x is the generator of the centre of the pure braid group of the stabiliser of x, and l denotes the length function on the braid group of G. Our proof is case-by-case using the Shephard–Todd classification of complex reflection groups. We also investigate the case where l(π ) − l(π x) − dn is as small as possible while still being positive, as well as a possible braid-theoretic explanation of our result.

This paper defines the partition algebra for complex reflection group $G(r,p,n)$ acting on $k$-fold tensor product $(\mathbb{C}^n)^{\otimes k}$, where $\mathbb{C}^n$ is the reflection representation of $G(r,p,n)$. A basis of the centralizer algebra of this action of $G(r,p,n)$ was given by Tanabe and for $p =1$, the corresponding partition algebra was studied by Orellana. We also establish a subalgebra as partition algebra of a subgroup of $G(r,p,n)$ acting on $(\mathbb{C}^n)^{\otimes k}$. We call these algebras as Tanabe algebras. The aim of this paper is to study representation theory of Tanabe algebras: parametrization of their irreducible modules, and construction of Bratteli diagram for the tower of Tanabe algebras. We conclude the paper by giving Jucys-Murphy elements of Tanabe algebras and their actions on the Gelfand-Tsetlin basis, determined by this multiplicity free tower, of irreducible modules.

Steinberg's fixed point theorem for crystallographic complex reflection groups

We review and introduce several approaches to the study of centralizer algebras of the infinite symmetric group $S_{\infty}$. Our work is led by the double commutant relationship between finite symmetric groups and partition algebras; in the case of $S_{\infty}$, we obtain centralizer algebras that are contained in partition algebras. In view of the theory of symmetric functions in non-commuting variables, we consider representations of $S_{\infty}$ that are faithful and that contain invariant elements; namely, non-unitary representations on sequence spaces. Nous étudions les algèbres du centralisateur du groupe symétrique infini $S_{\infty}$, passant en revue certaines approches et en introduisant de nouvelles. Notre travail est basé sur la relation du double commutant entre le groupe symétrique fini et les algèbres de partition; dans le cas de $S_{\infty}$, nous obtenons des algèbres du centralisateur contenues dans les algèbres de partition. Compte tenu de la théorie des fonctions symétriques en variables non commutatives, nous considérons les représentations de $S_{\infty}$ qui sont fidèles et contiennent les invariants; c’est-à-dire, les représentations non unitaires sur les espaces de suites.