arXiv subjects

Zhongguo Zhou

Publications and source records attributed to Zhongguo Zhou.

9 recordsLinked to original sources

arXiv2026

Three approaches to the Howe duality between quantum general linear supergroups

The Howe duality between quantum general linear supergroups was firstly established by Y. Zhang via quantum coordinate superalgebras. In this paper, we provide two other approaches to this Howe duality. One is constructed by quantum differential operators, while the other is based on the Beilinson-Lusztig-MacPherson realization of $U_q(\mathfrak{gl}_{m|n})$. Moreover, we show that these three approaches are equivalent by giving their action formulas explicitly.

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arXiv2020

Quantum queer supergroups via v-differential operators

By using certain quantum differential operators, we construct a super representation for the quantum queer supergroup U_v(q_n). The underlying space of this representation is a deformed polynomial superalgebra in 2n^2 variables whose homogeneous components can be used as the underlying spaces of queer q-Schur superalgebras. We then extend the representation to its formal power series algebra which contains a (super) submodule isomorphic to the regular representation of U_v(q_n). A monomial basis M for U_v(q_n) plays a key role in proving the isomorphism. In this way, we may present the quantum queer supergroup U_v(q_n) by another new basis L together with some explicit multiplication formulas by the generators. As an application, similar presentations are obtained for queer q-Schur superalgebras via the above mentioned homogeneous components. The existence of the bases M and L and the new presentation show that the seminal construction of quantum gl_n established by Beilinson-Lusztig-MacPherson thirty years ago extends to this "queer" quantum supergroup via a completely different approach.

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arXiv2019

The regular representation of $U_v(\mathfrak{gl}_{m|n})$

Using quantum differential operators, we construct a super representation of $U_v(\mathfrak{gl}_{m|n})$ on a certain polynomial superalgebra. We then extend the representation to its formal power series algebra which contains a $U_v(\mathfrak{gl}_{m|n})$-submodule isomorphic to the regular representation of $U_v(\mathfrak{gl}_{m|n})$. In this way, we obtain a presentation of $U_v(\mathfrak{gl}_{m|n})$ by a basis together with explicit multiplication formulas of the basis elements by generators.

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arXiv2018

Polynomial super representations of the hyperalgebra of $\mathfrak{gl}_{m|n}$ at roots of unity

As a homomorphic image of the hyperalgebra $U_{q,R}(m|n)$ associated with the quantum linear supergroup $U_\upsilon(\mathfrak{gl}_{m|n})$, we first give a presentation for the $q$-Schur superalgebra $S_{q,R}(m|n,r)$ over a commutative ring $R$. We then develop a criterion for polynomial supermodules of $U_{q,F}(m|n)$ over a filed $F$ and use this to determine a classification of polynomial irreducible supermodules at roots of unity. This also gives classifications of irreducible $S_{q,F}(m|n,r)$-supermodules for all $r$. As an application when $m=n\geq r$ and motivated by the beautiful work \cite{bru} in the classical (non-quantum) case, we provide a new proof for the Mullineux conjecture related to the irreducible modules over the Hecke algebra $H_{q^2,F}({\mathfrak S}_r)$; see \cite{Br} for a proof without using the super theory.

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arXiv2018

Multiplication formulas and semisimplicity for q-Schur superalgebras

We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for q-Schur superalgebras. This gives a combinatorialisation of the relative norm approach developed by the first two authors. We then give several applications of the multiplication formulas, including the matrix representation of the regular representation and a semisimplicity criterion for q-Schur superalgebras. We also construct infinitesimal and little q-Schur superalgebras directly from the multiplication formulas and develop their semisimplicity criteria.

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arXiv2012

Rigidity of formal characters of Lie algebras (II)

For a complex simple Lie algebra of type A_l,B_l,C_l or D_l, given a family of elements f_λ in commutative ring Z[Λ], we show that f_λ is just the formal character of the Weyl module V(λ) if f_λ satisfy several natural conditions. Hence we give a necessary and sufficient condition for constructing a family of g_l-modules from a family of g_{l-1}-modules.

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arXiv2012

Rigidity of formal character of Lie algebras of type A

For a complex simple Lie algebra of type $A$, given a family of elements $f_λ\in \mathbb Z[Λ],λ\in Λ^+,$ we show that $f_λ$ is just the formal character of the Weyl module $V(λ) $ if $f_λ$ satisfy two natural conditions.

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arXiv2012

Irreducible Characters for Algebraic Groups in Characteristic Three

In this note, we determine the irreducible characters for the simple algebraic groups of type $A_5$ over an algebraically closed field $K$ of characteristic 3, by using a theorem of Xi Nanhua and the Matlab software. In order to obtain higher speed than in Jiachen et al we modify the algorithm to compute the irreducible characters.

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arXiv2003

A New Multiplicity Formula for the Weyl Modules of Type A

A monomial basis and a filtration of subalgebras for the universal enveloping algebra $U(g_l)$ of a complex simple Lie algebra $g_l$ of type $A_l$ is given in this note. In particular, a new multiplicity formula for the Weyl module $V(λ)$ of $U(g_l)$ is obtained in this note.

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