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- 검색키워드
- 저자 : Benkart, G. (검색결과 5 건)
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Walks on graphs and their connections with tensor invariants and centralizer algebras
Benkart, Georgia,Moon, Dongho Elsevier 2018 Journal of algebra Vol.509 No.-
<P><B>Abstract</B></P> <P>The number of walks of <I>k</I> steps from the node 0 to the node <I>λ</I> on the McKay quiver determined by a finite group G and a G -module V is the multiplicity of the irreducible G -module <SUB> G λ </SUB> in the tensor power <SUP> V ⊗ k </SUP> , and it is also the dimension of the irreducible module labeled by <I>λ</I> for the centralizer algebra <SUB> Z k </SUB> ( G ) = <SUB> End G </SUB> ( <SUP> V ⊗ k </SUP> ) . This paper explores ways to effectively calculate that number using the character theory of G . We determine the corresponding Poincaré series. The special case λ = 0 gives the Poincaré series for the tensor invariants T <SUP> ( V ) G </SUP> = ⨁ k = 0 ∞ <SUP> ( <SUP> V ⊗ k </SUP> ) G </SUP> and a tensor analog of Molien's formula for polynomial invariants. When 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 McKay quivers, and the methods presented here provide efficient ways to compute the number of walks on them.</P>
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A Schur-Weyl Duality Approach toWalking on Cubes
Benkart, G.,Moon, D. Springer Science + Business Media 2016 Annals of Combinatorics Vol.20 No.3
<P>Walks on the representation graph (G) determined by a group G and a G-module V are related to the centralizer algebras of the action of G on the tensor powers via Schur-Weyl duality. This paper explores that connection when the group is and the module V is chosen so the representation graph is the n-cube. We describe a basis for the centralizer algebras in terms of labeled partition diagrams. We obtain an expression for the number of walks by counting certain partitions and determine the exponential generating functions for the number of walks.</P>
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Quantum walled Brauer-Clifford superalgebras
Benkart, G.,Guay, N.,Jung, J.H.,Kang, S.J.,Wilcox, S. Academic Press 2016 Journal of algebra Vol.454 No.-
<P>We introduce a new family of superalgebras, the quantum walled Brauer-Clifford superalgebras BCr,s (q). The superalgebra BCr,s (q) is a quantum deformation of the walled Brauer-Clifford superalgebra BCr,s and a super version of the quantum walled Brauer algebra. We prove that BCr,s (q) is the centralizer superalgebra of the action of U-q (q(n)) on the mixed tensor space V-q(r,s) = V-q(circle times r) circle times (V-q*)(circle times s) when n >= r + s, where V-q = C(q)((n\n)) is the natural representation of the quantum enveloping superalgebra U-q(q(n)) and V-q* is its dual space. We also provide a diagrammatic realization of BCr,s (q) as the (r, s)-bead tangle algebra BTr,s (q). Finally, we define the notion of q-Schur superalgebras of type Q and establish their basic properties. (C) 2015 Published by Elsevier Inc.</P>
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Benkart, Georgia,Kang, Seok-Jin,Oh, Se-jin,Park, Euiyong Oxford University Press 2014 International mathematics research notices Vol.2014 No.5
<P>We give an explicit construction of irreducible modules over Khovanov–Lauda–Rouquier algebras <I>R</I> and their cyclotomic quotients <I>R</I><SUP><I>λ</I></SUP> for finite classical types using a crystal basis theoretic approach. More precisely, for each element <I>v</I> of the crystal [Formula] (resp. <I>B</I>(<I>λ</I>)), we first construct certain modules <I>Δ</I>(<B>a</B>;<I>k</I>) labeled by the adapted string <B>a</B> of <I>v</I>. We then prove that the head of the induced module [Formula] is irreducible and that every irreducible <I>R</I>-module (resp. <I>R</I><SUP><I>λ</I></SUP>-module) can be realized as the irreducible head of one of the induced modules [Formula]. Moreover, we show that our construction is compatible with the crystal structure on [Formula] (resp. <I>B</I>(<I>λ</I>)).</P>
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