Celebratio Mathematica

G. Benkart, M. Chakrabarti, T. Halverson, R. Leduc, C. Lee, and J. Stroomer: “Tensor product representations of general linear groups and their connections with Brauer algebras,” J. Algebra 166 : 3 (June 1994), pp. 529–567. For J. Marshall Osborn and Louis Solomon on their 60th birthdays. MR 1280591 Zbl 0815.20028 article

For the complex general linear group \( G = GL(r,\mathbb{C}) \) we investigate the tensor product module \[\textstyle T = (\bigotimes^p V)\otimes(\bigotimes^q V^*) \] of \( p \) copies of its natural representation \( V = \mathbb{C}^r \) and \( q \) copies of the dual space \( V^* \) of \( V \). We describe the maximal vectors of \( T \) and from that obtain an explicit decomposition of \( T \) into its irreducible \( G \)-summands. Knowledge of the maximal vectors allows us to determine the centralizer algebra \( \mathscr{C} \) of all transformations on \( T \) commuting with the action of \( G \), to construct the irreducible \( \mathscr{C} \)-representations, and to identify \( \mathscr{C} \) with a certain subalgebra \( \mathscr{B}^{(r)}_{p,q} \) of the Brauer algebra \( \mathscr{B}(r)_{p+q} \).

Jeffrey D. Stroomer	Related
Thomas Michael Halverson	Related
Louis Solomon	Related
Robert Edgar Leduc	Related
Chanyoung Lee Shader	Related
Manish Chakrabarti	Related
J. Marshall Osborn	Related

G. Benkart, T. Halverson, and N. Harman: A parametric family of subalgebras of the Weyl algebra, III: Derivations. Preprint, October 2014. Part I was published in Trans. Am. Math. Soc. 367:3 (2015). Part II was published in Recent developments in algebraic and combinatorial aspects of representation theory (2013). ArXiv 1406.1508 techreport

An Ore extension over a polynomial algebra \( \mathbb{F}[x] \) is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra \( \mathrm{A}_h \) generated by elements \( x,y \), which satisfy \( yx-xy = h \), where \( h \) is in \( \mathbb{F}[x] \). When \( h \) is nonzero, these algebras are subalgebras of the Weyl algebra \( \mathrm{A}_1 \) and can be viewed as differential operators with polynomial coefficients. This paper determines the derivations of \( \mathrm{A}_h \) and the Lie structure of the first Hochschild cohomology group \[ \mathrm{HH}^1(\mathrm{A}_h) = \operatorname{Der}_{\mathbb{F}}(\mathrm{A}_h)/\operatorname{Inder}_{\mathbb{F}}(\mathrm{A}_h) \] of outer derivations over an arbitrary field. In characteristic 0, we show that \( \mathrm{HH}^1(\mathrm{A}_h) \) has a unique maximal nilpotent ideal modulo which it is 0 or a direct sum of simple Lie algebras that are field extensions of the one-variable Witt algebra. In positive characteristic we obtain decomposition theorems for \( \operatorname{Der}_{\mathbb{F}}(\mathrm{A}_h) \) and \( \mathrm{HH}^1(\mathrm{A}_h) \) and describe the structure of \( \mathrm{HH}^1(\mathrm{A}_h) \) as a module over the center of \( \mathrm{A}_h \).

Thomas Michael Halverson	Related
Nate Harman	Related

G. Benkart and T. Halverson: “Motzkin algebras,” European J. Combin. 36 (February 2014), pp. 473–502. MR 3131911 Zbl 1284.05333 ArXiv 1106.5277 article

We introduce an associative algebra \( \mathbf{M}_k(x) \) whose dimension is the \( 2k \)-th Motzkin number. The algebra \( \mathbf{M}_k(x) \) has a basis of “Motzkin diagrams”, which are analogous to Brauer and Temperley–Lieb diagrams. We show for a particular value of \( x \) that the algebra \( \mathbf{M}_k(x) \) is the centralizer algebra of the quantum enveloping algebra \( \mathbf{U}_{\mathbf{q}}(\mathfrak{gl}_2) \) acting on the \( k \)-fold tensor power of the sum of the 1-dimensional and 2-dimensional irreducible \( \mathbf{U}_{\mathbf{q}}(\mathfrak{gl}_2) \)-modules. We prove that \( \mathbf{M}_k(x) \) is cellular in the sense of Graham and Lehrer and construct indecomposable \( \mathbf{M}_k(x) \)-modules which are the left cell modules. When \( \mathbf{M}_k(x) \) is a semisimple algebra, these modules provide a complete set of representatives of isomorphism classes of irreducible \( \mathbf{M}_k(x) \)-modules. We compute the determinant of the Gram matrix of a bilinear form on the cell modules and use these determinants to show that \( \mathbf{M}_k(x) \) is semisimple exactly when \( x \) is not the root of certain Chebyshev polynomials.

G. Benkart, T. Halverson, and N. Harman: Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups. Preprint, May 2016. ArXiv 1605.06543 techreport

The partition algebra \( \mathrm{P}_k(n) \) and the symmetric group \( \mathrm{S}_n \) are in Schur–Weyl duality on the \( k \)-fold tensor power \( \mathrm{M}_n^{\otimes k} \) of the permutation module \( \mathrm{M}_n \) of \( \mathrm{S}_n \), so there is a surjection \[ \mathrm{P}_k(n)\to\mathrm{Z}_k(n) := \operatorname{End}_{\mathrm{S}_n}(\mathrm{M}_n^{\otimes k}) ,\] which is an isomorphism when \( n\neq 2k \). We prove a dimension formula for the irreducible modules of the centralizer algebra \( \mathrm{Z}_k(n) \) in terms of Stirling numbers of the second kind. Via Schur–Weyl duality, these dimensions equal the multiplicities of the irreducible \( \mathrm{S}_n \)-modules in \( \mathrm{M}_n^{\otimes k} \). Our dimension expressions hold for any \( n\geq 1 \) and \( k\geq 0 \). 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 \( \mathrm{M}_n^{\otimes k} \) and the quasi-partition algebra corresponding to tensor powers of the reflection representation of \( \mathrm{S}_n \).

Thomas Michael Halverson	Related
Nate Harman	Related

G. Benkart, T. Halverson, and N. Harman: Chip firing on Dynkin diagrams and McKay quivers. Preprint, February 2016. ArXiv 1601.06849 techreport

Two classes of avalanche-finite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chip-firing/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKay–Cartan matrices for finite subgroups \( \mathrm{G} \) of general linear groups. In the root system case, the recurrent and superstable configurations are identified explicitly and are related to minuscule dominant weights. In the McKay–Cartan case for finite subgroups of the special linear group, the cokernel is related to the abelianization of the subgroup \( \mathrm{G} \). In the special case of the classical McKay correspondence, the critical group and the abelianization are shown to be isomorphic.

Thomas Michael Halverson	Related
Nate Harman	Related

G. Benkart and T. Halverson: Walks on graphs and their connections with tensor invariants and centralizer algebras. Preprint, October 2016. ArXiv 1610.07837 techreport

The number of walks of \( k \) steps from the node \( \mathbf{0} \) to the node \( \lambda \) on the representation graph (McKay quiver) determined by a finite group \( \mathrm{G} \) and a \( \mathrm{G} \)-module \( \mathrm{V} \) is the multiplicity of the irreducible \( \mathrm{G} \)-module \( \mathrm{G}_{\lambda} \) in the tensor power \( \mathrm{V}^{\otimes k} \), and it is also the dimension of the irreducible module labeled by \( \lambda \) for the centralizer algebra \[ \mathrm{Z}_k(\mathrm{G}) = \operatorname{End}_{\mathrm{G}}(\mathrm{V}^{\otimes k}) .\] This paper explores ways to effectively calculate that number using the character theory of \( \mathrm{G} \). We determine the corresponding Poincaré series. The special case \( \lambda = 0 \) gives the Poincaré series for the tensor invariants \[ \mathrm{T}(\mathrm{V})^{\mathrm{G}} = \bigoplus_{k=0}^{\infty}(\mathrm{V}^{\otimes k})^{\mathrm{G}} .\] When \( \mathrm{G} \) is abelian, we show that the exponential generating function for the number of walks is a product of generalized hyperbolic functions. Many graphs (such as circulant graphs) can be viewed as representation graphs, and the methods presented here provide efficient ways to compute the number of walks on them.

J. M. Barnes, G. Benkart, and T. Halverson: “McKay centralizer algebras,” Proc. Lond. Math. Soc. (3) 112 : 2 (2016), pp. 375–414. MR 3471253 Zbl 1332.05147 ArXiv 1312.5254 article

For a finite subgroup \( \mathbf{G} \) of the special unitary group \( \mathrm{SU}_2 \), we study the centralizer algebra \[ Z_k(\mathbf{G}) = \operatorname{End}_{\mathbf{G}}(V^{\otimes k}) \] of \( \mathbf{G} \) acting on the \( k \)-fold tensor product of its defining representation \( \mathbf{V} = \mathbb{C}^2 \). These subgroups are in bijection with the simply laced affine Dynkin diagrams. The McKay correspondence relates the representation theory of these groups to the associated Dynkin diagram, and we use this connection to show that the structure and representation theory of \( \mathbf{Z}_k(\mathbb{G}) \) as a semisimple algebra is controlled by the combinatorics of the corresponding Dynkin diagram.

Jeffrey M. Barnes	Related
Thomas Michael Halverson	Related

G. Benkart and T. Halverson: Partition algebras \( \mathrm{P}_k(n) \) with \( 2k > n \) and the fundamental theorems of invariant theory for the symmetric group \( \mathrm{S}_n \). Preprint, July 2017. ArXiv 1707.01410 techreport

Assume \( \mathrm{M}_n \) is the \( n \)-dimensional permutation module for the symmetric group \( \mathrm{S}_n \), and let \( \mathrm{M}_n^{\otimes k} \) be its \( k \)-fold tensor power. The partition algebra \( \mathrm{P}_k(n) \) maps surjectively onto the centralizer algebra \( \operatorname{End}_{\mathrm{S}_n}(\mathrm{M}_n^{\otimes k}) \) for all \( k,n\in\mathbb{Z}_{\geq 1} \) and isomorphically when \( n\geq 2k \). We describe the image of the surjection \[ \Phi_{k,n}:\mathrm{P}_k(n)\to \operatorname{End}_{\mathrm{S}_n}(\mathrm{M}_n^{\otimes k}) \] explicitly in terms of the orbit basis of \( \mathrm{P}_k(n) \) and show that when \( 2k > n \) the kernel of \( \Phi_{k,n} \) is generated by a single essential idempotent \( e_{k,n} \), which is an orbit basis element. We obtain a presentation for \( \operatorname{End}_{\mathrm{S}_n}(\mathrm{M}_n^{\otimes k}) \) by imposing one additional relation, \( e_{k,n} = 0 \), to the standard presentation of the partition algebra \( \mathrm{P}_k(n) \) when \( 2k > n \). As a consequence, we obtain the fundamental theorems of invariant theory for the symmetric group \( \mathrm{S}_n \). We show under the natural embedding of the partition algebra \( \mathrm{P}_n(n) \) into \( \mathrm{P}_n(n) \) for \( k\geq n \) that the essential idempotent \( e_{n,n} \) generates the kernel of \( \Phi_{k,n} \). Therefore, the relation \( e_{n,n} = 0 \) can replace \( e_{k,n}=0 \) when \( k\geq n \).

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