arXiv subjects

Richard Dipper

Publications and source records attributed to Richard Dipper.

13 recordsLinked to original sources

arXiv2026

Coadjoint orbits of Row Closed Subgroups of $U_n(q)$

Let $n$ be a natural number, let $q$ be a prime power, and let $U_n(q)$ denote the group of unitriangular $n\times n$ matrices over the finite field $\mathbb{F}_q$ with $q$ elements. Row closed and column closed subgroups $U$ of $U_n(q)$ are special pattern subgroups obtained by deleting entire rows or, respectively, entire columns (apart from the diagonal entries). The supercharacters of the Andr\'{e}-Yan supercharacter theory of $U$ are afforded by the orbit modules arising from a monomial action of $U$ on the character group of the Lie algebra of $U$. We classify the corresponding orbits for row closed subgroups $U$, that is, we determine all orbits and identify which of the associated orbit modules are isomorphic and which afford orthogonal supercharacters. The classification for column closed subgroups follows from the mirror map, which reflects matrices across the antidiagonal.

math.RT↗

arXiv2023

Iwahori-Hecke algebras acting on tensor space by $q$-deformed letter permutations and $q$-partition algebras

Let $R$ be a commutative ring with identity and let $V$ be a free $R$-module of rank $n$ for some $n\in\mathbb{N}$. Fixing an $R$-basis $\mathcal{E}$ of $V$, the symmetric group $\mathfrak{S}_n$ acts on $V$ by permuting $\mathcal{E}$ and hence on tensor space $V^{\otimes r}$ for $r\in\mathbb{N}$ via the usual tensor product action turning $V$ and $V^{\otimes r}$ into $R\mathfrak{S}_n$-modules. For units $q$ in $R$ we construct an action of the corresponding Iwahori-Hecke algebra $\mathcal{H}_{R,q}(\mathfrak{S}_n)$ which specializes to the action of $R\mathfrak{S}_n$, if $q$ is taken to $1$. The centralizing algebra of this action is called the $q$-partition algebra $\mathcal{P}_{R,q}(n,r)$. Let $R$ be a field of characteristic not dividing $q$. We prove, that $\mathcal{P}_{R,q}(n,r)$ is isomorphic to the $q$-partition algebra defined by Halverson and Thiem by different means a few years ago.

math.RT↗

arXiv2018

On coadjoint orbits for $p$-Sylow subgroups of finite classical groups

Kirillov's orbit theory provides a powerful tool for the investigation of irreducible unitary representations of many classes of Lie groups. In a previous paper we used a modification hereof, called monomial linearisation, to construct a monomial basis of the regular representation of $p$-Sylow subgroups $U$ of the finite classical groups of untwisted type. In this sequel to this article we determine the stabilizers of special orbit generators and show, that for the groups of Lie type ${\mathfrak{B}}_n$ and ${\mathfrak{D}}_n$ a subclass of the orbit modules decompose the $U$-modules affording the Andr\'{e}-Neto supercharacters into a direct sum of submodules. Moreover these special orbit modules are either isomorphic or have no irreducible constituent in common, and each irreducible $U$ module is up to isomorphism constituent of precisely one of these.

math.RT↗

arXiv2017

Orbit method for $p$-Sylow subgroups of finite classical groups

For the $p$-Sylow subgroups $U$ of the finite classical groups of untwisted Lie type, $p$ an odd prime, we construct a monomial $\mathbb C U$-module $M$ which is isomorphic to the regular representation of $\mathbb C G$ by a modification of Kirillov's orbit method called monomial linearisation. We classify a certain subclass of orbits of the $U$-action on the monomial basis of $M$ consisting of so called staircase orbits and show, that every orbit module in $M$ is isomorphic to a staircase one. Finally we decompose the Andr\'e-Neto supercharacters of $U$ into a sum of $U$-characters afforded by staircase orbit modules contained in $M$.

math.RT↗

arXiv2017

On monomial linearisation and supercharacters of pattern subgroups

Column closed pattern subgroups $U$ of the finite upper unitriangular groups $U_n(q)$ are defined as sets of matrices in $U_n(q)$ having zeros in a prescribed set of columns besides the diagonal ones. We explain Jedlitschky's construction of monomial linearisation and apply this to $C U$ yielding a generalisation of Yan's coadjoint cluster representations. Then we give a complete classification of the resulting supercharacters, by describing the resulting orbits and determining the Hom-spaces between orbit modules.

math.RT↗

arXiv2014

$U_n(q)$ acting on flags and supercharacters

Let $U=U_n(q)$ be the group of lower unitriangular $n \times n$-matrices with entries in the field $\mathbb F_q$ with $q$ elements for some prime power $q$ and $n \in \mathbb N$. We investigate the restriction to $U$ of the permutation action of $GL_n(q)$ on flags in the natural $GL_n(q)$-module $\mathbb F_q^n$. Applying our results to the special case of flags of length two we obtain a complete decomposition of the permutation representation of $GL_n(q)$ on the cosets of maximal parabolic subgroups into irreducible $\mathbb C U$-modules.

math.RT↗

arXiv2013

Irreducible constituents of minimal degree in supercharacters of the finite unitriangular groups

Let $q$ be a prime power and $U$ the group of lower unitriangular matrices of order $n$ for some natural number $n$. We give a lower bound for the degrees of irreducible constituents of André-Yan supercharacters and classify the supercharacters having constituents whose degree assume this lower bound. Moreover we show that the number of distinct irreducible characters of $U$ meeting this condition is a polynomial in $(q-1)$ with nonnegative integral coefficients and exhibit monomial sources for those.

math.RT↗

arXiv2012

Symmetrizers and antisymmetrizers for the BMW algebra

Let $n\in\mathds{N}$ and $B_n(r,q)$ be the generic Birman-Murakami-Wenzl algebra with respect to indeterminants $r$ and $q$. It is known that $B_n(r,q)$ has two distinct linear representations generated by two central elements of $B_n(r,q)$ called the symmetrizer and antisymmetrizer of $B_n(r,q)$. These generate for $n\geq 3$ the only one dimensional two sided ideals of $B_n(r,q)$ and generalize the corresponding notion for Hecke algebras of type $A$. The main result in this paper explicitly determines the coefficients of these elements with respect to the graphical basis of $B_n(r,q)$.

math.QA↗

arXiv2007

The rational Schur algebra

We extend the family of classical Schur algebras in type A, which determine the polynomial representation theory of general linear groups over an infinite field, to a larger family, the rational Schur algebras, which determine the rational representation theory of general linear groups over an infinite field. This makes it possible to study the rational representation theory of such general linear groups directly through finite dimensional algebras. We show that rational Schur algebras are quasihereditary over any field, and thus have finite global dimension. We obtain explicit cellular bases of a rational Schur algebra by a descent from a certain ordinary Schur algebra. We also obtain a description, by generators and relations, of the rational Schur algebras in characteristic zero.

math.RT↗

arXiv2005

Brauer algebras, symplectic Schur algebras and Schur-Weyl duality

In this paper we prove Schur-Weyl duality between the symplectic group and Brauer algebra over an arbitrary infinite field $K$. We show that the natural homomorphism from the Brauer algebra $B_n(-2m)$ to the endomorphism algebra of tensor space $(K^{2m})^{\otimes n}$ as a module over the symplectic similitude group $GSp_{2m}(K)$ (or equivalently, as a module over the symplectic group $Sp_{2m}(K)$) is always surjective. Another surjectivity, that of the natural homomorphism from the group algebra for $GSp_{2m}(K)$ to the endomorphism algebra of $(K^{2m})^{\otimes n}$ as a module over $B_n(-2m)$, is derived as an easy consequence of S.~Oehms' results [S. Oehms, J. Algebra (1) 244 (2001), 19--44].

math.RT↗

arXiv2000

Morita equivalences of Ariki-Koike algebras

We prove that every Ariki-Koike algebra is Morita equivalent to a direct sum of tensor products of smaller Ariki-Koike algebras which have q-connected parameter sets. A similar result is proved for the cyclotomic q-Schur algebras. Combining our results with work of Ariki and Uglov, the decomposition numbers for the Ariki-Koike algebras defined over fields of characteristic zero are now known in principle.

math.RT↗

arXiv1998

Generalized q-Schur algebras and modular representation theory of finite groups with split (BN)-pairs

We introduce a generalized version of a q-Schur algebra (of parabolic type) for arbitrary Hecke algebras over extended Weyl groups. We describe how the decomposition matrix of a finite group with split BN-pair, with respect to a non-describing prime, can be partially described by the decomposition matrices of suitably chosen q-Schur algebras. We show that the investigated structures occur naturally in finite groups of Lie type.

math.QA↗

arXiv1997

The (Q,q)-Schur algebra

In this paper we use the Hecke algebra of type $B$ to define a new algebra $\Sch$ which is an analogue of the q-Schur algebra. We construct Weyl modules for $\Sch$ and obtain, as factor modules, a family of irreducible $\Sch$-modules over any field.

q-alg↗