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Deke Zhao

Publications and source records attributed to Deke Zhao.

At least 19 recordsLinked to original sources
arXiv2026

Remarks on the shuffle elements of Iwahori--Hecke algebras

Let $H_n(q)$ be the generic Iwahori--Hecke algebra associated to the symmetric group $\mathfrak{S}_n$ of degree $n$ and let $\mathscr{Y}_{n,i}$ be the sum of standard basis of $H_n(q)$ with Coxeter length $i$ for $1\leq i\leq \frac{n(n-1)}{2}$. We show that $\mathscr{Y}_{n,1}$ and $\mathscr{Y}_{n,2}$ are noncommutative whenever $n\geq 4$, which disproves Doikou's Conjecture on the commutativity of the shuffle elements of $H_n(q)$ in (Nuclear Physics B 1029 (2026): 117532). We also show that the matrix of the operator of the left multiplication by $\mathscr{Y}_{n,i}$ in $H_n(q)$ is $q$-symmetric for all $i$.

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arXiv2026

A categorification of the Brenti--Welker identity

The paper aims to provide a categorification of the Brenti--Welker identity involving Eulerian numbers in (Adv. Appl. Math. 42 (2009): 545--556) by lifting it from an enumerative equality to an isomorphism of symmetric group representations. To do so, we study the decomposition of the tensor product of $(\mathbb{C}^r)^{\otimes n}$ and modules affording Foulkes characters as modules of the symmetric group. The main ingredient of the proof is a combinatorial identity which may be of independent interest.

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arXiv2024

Revisiting Foulkes characters of wreath products

The article is concerned with the Foulkes characters of wreath products, which are block characters of wreath products, i.e., the positive-definite class functions depending only on the length of its elements. Inspired by the works of Gnedin--Gorin--Kerov and Miller, we introduce two specializations of the Schur--Weyl--Sergeev duality for wreath products and obtain two families of block characters, which provide a decomposition and an alternative construction of the Foulkes characters of wreath products. In particular, we give alternative proofs on some remarkable properties of the Foulkes characters. Along the way, we show that the Foulkes characters are the extreme rays of the cone of the block characters of wreath products and construct the representations with traces being the Foulkes characters via the coinvariant algebra of wreath products.

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arXiv2024

Characters of Ariki--Koike algebras

In this paper, we prove the Regev formulae for the characters of the Ariki--Koike algebras by applying the Schur--Sergeev reciprocity between the quantum superalgebras and the Ariki--Koike algebras, which is a generalization of the formulas in (D. Zhao, Israel J. Math. 229 (2019): 67--83 ). As a corollary, we provide the Regev formulae for the characters of the complex reflection group of type $G(m,1,n)$, which is a generalization of the formulas in (A. Regev, Israel J. Math. 195 (2013): 31--35).

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arXiv2024

Markov traces on degenerate cyclotomic Hecke algebras

Let $H_n(\boldsymbol{u})$ be the degenerate cyclotomic Hecke algebra with parameter $\boldsymbol{u}=(u_1, \ldots, u_m)$ over $\mathbb{C}(\boldsymbol{u})$. We define and construct the (non-)normalized Markov traces on the sequence $\{H_n(\boldsymbol{u})\}_{n=1}^{\infty}$. This allows us to provide a canonical symmetrizing form on $H_n(\boldsymbol{u})$ and show that the Brudan--Kleshchev trace on $H_n(\boldsymbol{u})$ is a specialization of the non-normalized Markov traces.

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arXiv2022

Schur-Sergeev duality for Ariki-Koike algebras

Let $U_q(\mathfrak{g})$ be the quantized superalgebra of $\mathfrak{g}=\mathfrak{gl}(k_1|\ell_1)\oplus\cdots\oplus\mathfrak{gl}(k_m|\ell_m)$ and $H_{m,n}(q,\mathbf{Q})$ the cyclotomic Hecke algebra of type $G(m,1,n)$. We define a right $H_{m,n}(q,\mathbf{Q})$-action on the $n$-fold tensor (super)space of the vector representation of $U_q(\mathfrak{g})$ and prove the Schur--Weyl reciprocity between $U_q(\mathfak{g})$ and $H_{m,n}(q,\mathbf{Q})$.

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arXiv2022

Cyclotomic $q$-Schur superalgebras

The paper aims to introduce the cyclotomic $q$-Schur superalgebra via the permutation supermodules of the cyclotomic Hekce algebra and investigate its structure. In particular, we show that the cyclotomic $q$-Schur superalgebra is a cellular superalgebra and establish the Schur-Weyl duality between the cyclotomic Hecke algebra and the cyclotomic $q$-Schur superalgebra.

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arXiv2019

A super Frobenius formula for the characters of cyclotomic Hecke algebras

We prove a super Frobenius formula for the characters of the cyclotomic Hecke algebras by applying the super Schur-Weyl reciprocity between the quantum superalgebras and cyclotomic Hecke algebras, which is a super analogue of the Frobenius formula in (T. Shoji, J. Algebra 226: 2000, 818--856) and a cyclotomic analogue of the super Frobeinus formula in (H. Mitsuhashi, Linear Multilinear Algebra 58: 2010, 941--955).

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arXiv2019

RSK superinsertion and super Frobenius formulae

In this paper we extend the Robinson-Schensted-Knuth (RSK) superinsertion algorithm to hook-multipartitions and derive the super Frobenius formula for the characters of cyclotomic Hecke algebras (Linear and Multilinear Algebra, DOI: 10.1080/03081087.2019.1663140) via the RSK superinsertion algorithm. In particular, we obtain a new proof of Mitsuhashi's super Frobenius formula for the characters of Iwahori-Hecke algebras.

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arXiv2018

Characters of Iwahori-Hecke algebras

In this paper we prove a quantum generalization of Regev's theorems in (Israel. J. Math. 195 (2013), 31--35) by applying the Schur-Weyl duality between the quantum superalgebra and Iwahori-Hecke algebra. We also present an alternative proof of the quantized generalizations using the skew character theory of Iwahori-Hecke algebras.

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arXiv2017

Notes on graded symmetric cellular algebras

Let $A=\oplus_{i\in \mathbb{Z}}A_i$ be a finite dimensional graded symmetric cellular algebra with a homogeneous symmetrizing trace of degree $d$. We prove that $A_{-d}$ contains the Higman ideal $H(A)$ of the center of $A$ and $\dim H(A)\leq \dim A_{0}$ if $d\neq 0$, and provide a semisimplicity criterion of $A$ in terms of the centralizer of $A_0$, which is a graded version of \cite[Theorem 3.2]{L}.

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arXiv2013

Fusion procedure for Degenerate cyclotomic Hecke algebras

The primitive idempotents of the generic degenerate cycloctomic Hecke algebras are derived by consecutive evaluations of a certain rational function. This rational function depends only on the Specht modules and the normalization factors are the weights of the Brundan-Kleshchev trace.

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arXiv2013

Hochschild and cyclic (co)homology of superadditive categories

We define the Hochschild and cyclic (co)homology groups for superadditive categories and show that these (co)homology groups are graded Morita invariants. We also show that the Hochschild and cyclic homology are compatible with the tensor product of superadditive categories.

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arXiv2013

The Symbolic and cancellation-free formulae for Schur elements

In this paper we give a symbolical formula and a cancellation-free formula for the Schur elements associated to the simple modules of the degenerate cyclotomic Hecke algebras. As some direct applications, we show that the Schur elements are symmetric with respect to the natural symmetric group action and are integral coefficients polynomials and we give a different proof of Ariki-Mathas-Rui's criterion on the semi-simplicity of degenerate cyclotomic Hecke algebras.

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arXiv2013

Projective cell modules of Frobenius cellular algebras

For a finite dimensional Frobenius cellular algebra, a sufficient and necessary condition for a simple cell module to be projective is given. A special case that dual bases of the cellular basis satisfying a certain condition is also considered. The result is similar to that in symmetric case.

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arXiv2012

Graded Morita equivalence of Clifford superalgebras

This note uses a variation of graded Morita theory for finite dimensional superalgebras to determine explicitly the graded basic superalgebras for all real and complex Clifford superalgebras. As an application, the Grothendieck groups of the category of left $\mathbb{Z}_2$-graded modules over all real and complex Clifford superalgebras are described explicitly.

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arXiv2011

Matrix units and Schur elements for the degenerate cyclotomic Hecke algebras

The paper uses the cellular basis of the (semi-simple) degenerate cyclotomic Hecke algebras to investigate these algebras exhaustively. As a consequence, we describe explicitly the "Young's seminormal form" and a orthogonal bases for Specht modules and determine explicitly the closed formula for the natural bilinear form on Specht modules and Schur elements for the degenerate cyclotomic Hekce algebras.

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