Searcharxiv⌕ Search

arXiv subjects

Hebing Rui

Publications and source records attributed to Hebing Rui.

29 records · Page 2Linked to original sources
arXiv2014

Highest weight vectors of mixed tensor products of general linear Lie superalgebras

In this paper, a notion of cyclotomic (or level $k$) walled Brauer algebras $\mathscr B_{k, r, t}$ is introduced for arbitrary positive integer $k$. It is proven that $\mathscr B_{k, r, t}$ is free over a commutative ring with rank $k^{r+t}(r+t)!$ if and only if it is admissible. Using super Schur-Weyl duality between general linear Lie superalgebras $\mathfrak{gl}_{m|n}$ and $\mathscr B_{2, r, t}$, we give a classification of highest weight vectors of $\mathfrak{gl}_{m|n}$-modules $M_{pq}^{rt}$, the tensor products of Kac-modules with mixed tensor products of the natural module and its dual. This enables us to establish an explicit relationship between $\mathfrak{gl}_{m|n}$-Kac-modules and right cell (or standard) $\mathscr B_{2, r, t}$-modules over $\mathbb C$. Further, we find an explicit relationship between indecomposable tilting $\mathfrak{gl}_{m|n}$-modules appearing in $M_{pq}^{rt}$, and principal indecomposable right $\mathscr B_{2, r, t}$-modules via the notion of Kleshchev bipartitions. As an application, decomposition numbers of $\mathscr B_{2, r, t}$ arising from super Schur-Weyl duality are determined.

math.QA↗
arXiv2013

Affine walled Brauer algebras

A new class of associative algebras referred to as affine walled Brauer algebras are introduced. These algebras are free with infinite rank over a commutative ring containing 1. Then level two walled Brauer algebras over C are defined, which are some cyclotomic quotients of affine walled Brauer algebras. We establish a super Schur-Weyl duality between affine walled Brauer algebras and general linear Lie superalgebras, and realize level two walled Brauer algebras as endomorphism algebras of tensor modules of Kac modules with mixed tensor products of the natural module and its dual over general linear Lie superalgebras, under some conditions. We also prove the weakly cellularity of level two walled Brauer algebras, and give a classification of their irreducible modules over C. This in tur enables us to classify the indecomposable direct summands of the said tensor modules.

math.RT↗
arXiv2011

Singular parameters for the Birman-Murakami-Wenzl algebra

In this paper, we classify the singular parameters for the Birman-Murakami-Wenzl algebra over an arbitrary field. Equivalently, we give a criterion for the Birman-Murakami-Wenzl algebra being Morita equivalent to the direct sum of the Hecke algebras associated to certain symmetric groups.

math.QA↗
arXiv2010

Quantum Schur Superalgebras and Kazhdan-Lusztig Combinatorics

We introduce the notion of quantum Schur (or $q$-Schur) superalgebras. These algebras share certain nice properties with $q$-Schur algebras such as base change property, existence of canonical $\mathbb Z[v,v^{-1}]$-bases, and the duality relation with quantum matrix superalgebra $\sA(m|n)$. We also construct a cellular $\mathbb Q(\up)$-basis and determine its associated cells, called super-cells, in terms of a Robinson--Schensted--Knuth super-correspondence. In this way, we classify all irreducible representations over $\mathbb Q(\up)$ via super-cell modules.

math.QA↗
arXiv2007

Discriminants of Brauer algebras

In this paper, we compute the discriminant of the Gram matrix associated to each cell module of the Brauer algebra $\cba{n}$. Theoretically, we know when a cell module of $\cba{n}$ is equal to its simple head. This gives a solution of this long standing problem.

math.QA↗
arXiv2006

Cyclotomic Nazarov-Wenzl algebras

Nazarov \cite{Nazarov:brauer} introduced an infinite dimensional algebra, which he called the \textit{affine Wenzl algebra}, in his study of the Brauer algebras. In this paper we study certain ``cyclotomic quotients'' of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank $r^n(2n-1)!!$ (when $Ω$ is $\bu$--admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov--Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Weyl algebra (when $Ω$ is admissible).

math.QA↗
arXiv1999

Ariki-Koike Algebras with Semisimple Bottoms

We investigated the representation thoery of an Ariki-Koike algebra whose Poincare polynomial associated with the "bottom", i.e., the subgroup on which the symmetric group acts, is non-zero in the base field. We proved that the module category of such an Ariki-Koike algebra is Morita equivalent to the module category of a direct sum of tensor products of Hecke algebras associated with certain symmetric groups. We also generalized this Morita equivalence theorem to give a Morita equivalenve between a $q$-Schur$^m$ algebra and a direct sum of tensor products of certain $q$-Schur algebras.

math.QA↗