Celebratio Mathematica

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 \( \mathrm{A}_h \).

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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 \( \mathrm{S}_n \).

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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

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