arXiv subjects

Rudolf Tange

Publications and source records attributed to Rudolf Tange.

At least 19 recordsLinked to original sources

arXiv2024

A Frobenius splitting and cohomology vanishing for the cotangent bundles of the flag varieties of GL_n

Let k be an algebraically closed field of characteristic p>0, let G=GL_n be the general linear group over k, let P be a parabolic subgroup of G, and let u_P be the Lie algebra of its unipotent radical. We show that the Kumar-Lauritzen-Thomsen splitting of the cotangent bundle Gx^Pu_P of G/P has top degree (p-1)\dim(G/P). The component of that degree is therefore given by the (p-1)-th power of a function f. We give a formula for f and deduce that it vanishes on the exceptional locus of the resolution Gx^Pu_P-->\ov{\mc O} where \ov{\mc O} is the closure of the Richardson orbit of P. As a consequence we obtain that the higher cohomology groups of a line bundle on Gx^Pu_P associated to a dominant weight are zero. The splitting of Gx^Pu_P given by f^{p-1} can be seen as a generalisation of the Mehta-Van der Kallen splitting of Gx^Bu.

math.AG↗

arXiv2023

Invariants in divided power algebras

Let $k$ be an algebraically closed field of characteristic $p>0$, let G=GL_n be the general linear group over $k$, let g=gl_n be its Lie algebra and let $D_s$ be subalgebra of the divided power algebra of g^* spanned by the divided power monomials with exponents $ 1$. If $s=1$, then $D_1$ is isomorphic to the truncated coordinate ring of g of dimension p^{dim(g)} and we conjecture that the restriction property holds and show that this leads to a conjectural spanning set for the invariants (in all degrees). We give similar results for the divided power algebras of several matrices and of vectors and covectors, and show that in the second case the restriction property doesn't hold. We also give the dimensions of the filtration subspaces of degree $\le n$ of the centre of the hyper algebra of Dist(G_s).

math.RT↗

arXiv2023

A combinatorial translation principle and diagram combinatorics for the symplectic group

Let k be an algebraically closed field of characteristic p>2. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the symplectic group over k in terms of cap-curl diagrams under the assumption that p is bigger than the greatest hook length in the largest partition involved. As a corollary we obtain the decomposition numbers for the Brauer algebra under the same assumptions. Our work combines ideas from work of Cox and De Visscher and work of Shalile with techniques from the representation theory of reductive groups.

math.RT↗

arXiv2023

A combinatorial translation principle and diagram combinatorics for the general linear group

Let k be an algebraically closed field of characteristic p>0. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the general linear group over k in terms of cap diagrams under the assumption that p is bigger than the greatest hook length in the partitions involved. Then we introduce and study the rational Schur functor from a category of GL_n-modules to the category of modules for the walled Brauer algebra. As a corollary we obtain the decomposition numbers for the walled Brauer algebra when p is bigger than the greatest hook length in the partitions involved. This is a sequel to an earlier paper on the symplectic group and the Brauer algebra.

math.RT↗

arXiv2023

Injective and tilting resolutions and a Kazhdan-Lusztig theory for the general linear and symplectic group

Let k be an algebraically closed field of characteristic p>0 and let G be a symplectic or general linear group over k. We consider induced modules for G under the assumption that p is bigger than the greatest hook length in the partitions involved. We give explicit constructions of left resolutions of induced modules by tilting modules. Furthermore, we give injective resolutions for induced modules in certain truncated categories. We show that the multiplicities of the indecomposable tilting and injective modules in these resolutions are the coefficients of certain Kazhdan-Lusztig polynomials. We also show that our truncated categories have a Kazhdan-Lusztig theory in the sense of Cline, Parshall and Scott. This builds further on work of Cox-De Visscher and Brundan-Stroppel.

math.RT↗

arXiv2018

On the first restricted cohomology of a reductive Lie algebra and its Borel subalgebras

Let k be an algebraically closed field of characteristic p>0 and let G be a connected reductive group over k. Let B be a Borel subgroup of G and let g and b be the Lie algebras of G and B. Denote the first Frobenius kernels of G and B by G_1 and B_1. Furthermore, denote the algebras of polynomial functions on G and g by k[G] and k[g], and similar for B and b. The group G acts on k[G] via the conjugation action and on k[g] via the adjoint action. Similarly, B acts on k[B] via the conjugation action and on k[b] via the adjoint action. We show that, under certain mild assumptions, the cohomology groups H^1(G_1,k[g]), H^1(B_1,k[b]), H^1(G_1,k[G]) and H^1(B_1,k[B]) are zero. We also extend all our results to the cohomology for the higher Frobenius kernels.

math.RT↗

arXiv2018

Bases for spaces of highest weight vectors in arbitrary characteristic

Let k be an algebraically closed field of arbitrary characteristic. First we give explicit bases for the highest weight vectors for the action of GL_r x GL_s on the coordinate ring k[Mat_{rs}^m] of m-tuples of r x s-matrices. It turns out that this is done most conveniently by giving an explicit good GL_r x GL_s-filtration on k[Mat_{rs}^m]. Then we deduce from this result explicit spanning sets of the k[Mat_n]^{GL_n}-modules of highest weight vectors in the coordinate ring k[Mat_n] under the conjugation action of GL_n.

math.RT↗

arXiv2017

Highest weight vectors and transmutation

Let $G={\rm GL}_n$ be the general linear group over an algebraically closed field $k$, let $\mathfrak g=\mathfrak gl_n$ be its Lie algebra and let $U$ be the subgroup of $G$ which consists of the upper uni-triangular matrices. Let $k[\mathfrak g]$ be the algebra of polynomial functions on $\mathfrak g$ and let $k[\mathfrak g]^G$ be the algebra of invariants under the conjugation action of $G$. In characteristic zero, we give for all dominant weights $χ\in\mathbb Z^n$ finite homogeneous spanning sets for the $k[\mathfrak g]^G$-modules $k[\mathfrak g]_χ^U$ of highest weight vectors. This result (with some mistakes) was already given without proof by J.~F.~Donin. Then we do the same for tuples of $n\times n$-matrices under the diagonal conjugation action. Furthermore we extend our earlier results in positive characteristic and give a general result which reduces the problem to giving spanning sets of the highest weight vectors for the action of ${\rm GL}_r\times{\rm GL}_s$ on tuples of $r\times s$ matrices. This requires the technique called "transmutation" by R.~Brylinsky which is based on an instance of Howe duality. In the cases that $χ_{{}_n}\ge -1$ or $χ_{{}_1}\le 1$ this leads to new spanning sets for the modules $k[\mathfrak g]_χ^U$.

math.RT↗

arXiv2017

Embeddings of spherical homogeneous spaces in characteristic p

Let G be a reductive group over an algebraically closed field of characteristic p>0. We study properties of embeddings of spherical homogeneous G-spaces. We look at Frobenius splittings, canonical or by a (p-1)-th power, compatible with certain subvarieties. We also look at cohomology vanishing and show the existence of rational G-equivariant resolutions by toroidal embeddings. We show that the class of homogeneous spaces for which our results hold contains the symmetric homogeneous spaces in characteristic not 2 and is closed under parabolic induction.

math.AG↗

arXiv2014

Highest-weight vectors for the adjoint action of GL_n on polynomials, II

Let G=GL_n be the general linear group over an algebraically closed field k and let g=gl_n be its Lie algebra. Let U be the subgroup of G which consists of the upper unitriangular matrices. Let k[g] be the algebra of polynomial functions on g and let k[g]^G be the algebra of invariants under the conjugation action of G. For all weights chi in Z^n with chi_2<=0 or chi_{n-1}>=0 we give explicit bases for the k[g]^G-module k[g]^U_chi of highest weight vectors of weight chi. This extends earlier results to a much bigger class of weights. To express our semi-invariants in terms of matrix powers we prove certain Cayley-Hamilton type identities.

math.RT↗

arXiv2012

On embeddings of certain spherical homogeneous spaces in prime characteristic

Let $\mc G$ be a reductive group over an algebraically closed field of characteristic $p>0$. We study homogeneous $\mc G$-spaces that are induced from the $G\times G$-space $G$, $G$ a suitable reductive group, along a parabolic subgroup of $\mc G$. We show that, under certain mild assumptions, any (normal) equivariant embedding of such a homogeneous space is canonically Frobenius split compatible with certain subvarieties and has an equivariant rational resolution by a toroidal embedding. In particular, all these embeddings are Cohen-Macaulay. Examples are the $G\times G$-orbits in normal reductive monoids with unit group $G$. Our class of homogeneous spaces also includes the open orbits of the well-known determinantal varieties and the varieties of (circular) complexes. We also show that all $G$-orbit closures in a spherical variety which is canonically Frobenius split are normal. Finally we study the Gorenstein property for the varieties of circular complexes and for a related reductive monoid.

math.AG↗

arXiv2012

Highest weight vectors for the adjoint action of GL_n on polynomials

Let G=GL_n be the general linear group over an algebraically closed field k and let g=gl_n be its Lie algebra. Let U be the subgroup of G which consists of the upper unitriangular matrices. Let k[g] be the algebra of regular functions on $\g$. For 2(n-1)-1 weights we give explicit bases for the k[g]^G-module k[g]^U_λof highest weight vectors of weight λ. For 5 of those weights we show that this basis is algebraically independent over the invariants k[g]^G and generates the k[g]^G-algebra $\bigoplus_{r\ge0}k[\g]^U_{rλ}$. Finally we formulate a question which asks whether in characteristic zero k[g]^G-module generators of k[g]^U_λcan be obtained by applying one explicit highest weight vector of weight λin the tensor algebra T(g) to varying tuples of fundamental invariants.

math.RT↗

arXiv2011

Factorisation properties of group scheme actions

Let H be an algebraic group scheme over a field k acting on a commutative k-algebra A which is a unique factorisation domain. We show that, under certain mild assumptions, the monoid of nonzero H-stable principal ideals in A is free commutative. From this we deduce, in certain special cases, results about the monoid of nonzero semi-invariants and the algebra of invariants.

math.AC↗

arXiv2010

A bideterminant basis for a reductive monoid

We use the rational tableaux introduced by Stembridge to give a bideterminant basis for a normal reductive monoid and for its variety of noninvertible elements. We also obtain a bideterminant basis for the full coordinate ring of the general linear group and for all its truncations with respect to saturated sets. Finally, we deduce an alternative proof of the double centraliser theorem for the rational Schur algebra and the walled Brauer algebra over an arbitrary infinite base field which was first obtained by Dipper, Doty and Stoll.

math.RT↗

arXiv2009

The Zassenhaus variety of a reductive Lie algebra in positive characteristic

Let g be the Lie algebra of a connected reductive group G over an algebraically closed field k of characteristic p>0. Let $Z$ be the centre of the universal enveloping algebra U=U(g) of g. Its maximal spectrum is called the Zassenhaus variety of g. We show that, under certain mild assumptions on G, the field of fractions Frac(Z) of Z is G-equivariantly isomorphic to the function field of the dual space g* with twisted G-action. In particular Frac(Z) is rational. This confirms a conjecture J. Alev. Furthermore we show that Z is a unique factorisation domain, confirming a conjecture of A. Braun and C. Hajarnavis. Recently, A. Premet used the above result about Frac(Z), a result of Colliot-Thelene, Kunyavskii, Popov and Reichstein and reduction mod p arguments to show that the Gelfand-Kirillov conjecture cannot hold for simple complex Lie algebras that are not of type A, C or G_2.

math.RA↗

arXiv2009

The Brauer algebra and the symplectic Schur algebra

Let k be an algebraically closed field of characteristic p>0, let m,r be integers with m\ge1, r\ge0 and m\ge r and let S_0(2m,r) be the symplectic Schur algebra over k as introduced by the first author. We introduce the symplectic Schur functor, derive some basic properties of it and relate this to work of Hartmann and Paget. We do the same for the orthogonal Schur algebra. We give a modified Jantzen sum formula and a block result for the symplectic Schur algebra under the assumption that r and the residue of 2m mod p are small relative to p. From this we deduce a block result for the orthogonal Schur algebra under similar assumptions. Finally, we deduce from the previous results a new proof of the geometric description of the blocks of the Brauer algebra in characteristic 0 as obtained by Cox, De Visscher and Martin.

math.RT↗

arXiv2008

Complete Reducibility and Separability

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre's concept of G-complete reducibility for subgroups of G. The separability hypothesis appears in many general theorems concerning G-complete reducibility. We demonstrate that many of these results fail without this hypothesis. On the other hand, we prove that if G is a connected reductive group and p is very good for G, then any subgroup of G is separable; we deduce that under these hypotheses on G, a subgroup H of G is G-completely reducible provided the Lie algebra of G is semisimple as an H-module. Recently, Guralnick has proved that if H is a reductive subgroup of G and C is a conjugacy class of G, then the intersection of C and H is a finite union of H-conjugacy classes. For generic p -- when certain extra hypotheses hold, including separability -- this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztig's deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick's result is false if one considers conjugacy classes of n-tuples of elements from H for n > 1.

math.GR↗

arXiv2007

The symplectic ideal and a double centraliser theorem

We interpret a result of S. Oehms as a statement about the symplectic ideal. We use this result to prove a double centraliser theorem for the symplectic group acting on \bigoplus_{r=0}^s\otimes^rV, where V is the natural module for the symplectic group. This result was obtained in characteristic zero by H. Weyl. Furthermore we use this to extend to arbitrary connected reductive groups G with simply connected derived group the earlier result of the author that the algebra K[G]^g of infinitesimal invariants in the algebra of regular functions on G is a unique factorisation domain.

math.AC↗