Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.14638

The Cartan determinant conjecture for representation-finite algebras over perfect fields

Liang Chen
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $k$ be a perfect field and let $A$ be a finite-dimensional representation-finite $k$-algebra. We prove that finite global dimension implies that the integral Cartan determinant of $A$ is one, extending Enomoto's result from algebraically closed fields to perfect fields.

Explore related subjects

math.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Liang Chen. 2026-10-10. The Cartan determinant conjecture for representation-finite algebras over perfect fields. https://arxiv.org/abs/2610.14638

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Middle convolution for Lie algebra representations

This paper introduces a Lie algebra analogue of the middle convolution functor, which is defined on the category of modules over certain Lie algebras, including, as typical motivating examples, free Lie algebras, Drinfeld-Kohno Lie algebras, and the holonomy Lie algebras of complements of hyperplane arrangements. First, we demonstrate that the middle convolution for Lie algebra representations can be regarded as a natural generalization of the infinitesimal analogue of the Long-Moody functor for Drinfeld-Kohno Lie algebras. Second, we show that our middle convolution recovers the Dettweiler-Reiter additive middle convolution for Fuchsian systems on the punctured Riemann sphere as a special case. Furthermore, we show that when applied to the holonomy Lie algebra of the complement of a hyperplane arrangement, our functor is compatible with Haraoka's middle convolution for logarithmic connections on such complements. Finally, we establish a Riemann-Hilbert correspondence between the middle convolution for the holonomy Lie algebra and the middle convolution for local systems on complements of hyperplane arrangements.

math.RT↗
arXiv2026

Gelfand--Kirillov dimensions of highest weight modules for basic classical Lie superalgebras

We give exact Gelfand--Kirillov (GK) dimension formulas for simple highest weight modules of the basic classical Lie superalgebras considered in this paper. For the type II orthosymplectic algebras $D(m,n)$ and $B(m,n)$, we identify the PBW associated variety as a product of factor supports evaluated at highest weights for two compatible Borel subalgebras. This gives algorithms for arbitrary complex highest weights, without integrality or typicality hypotheses. In type $B$, the terminal factor is $\mathfrak{osp}(1|2n)$, for which we derive an insertion formula that distinguishes integer and half-integer shifted coordinates. For $G(3)$, $F(4)$ and $D(2,1;α)$, finite odd-reflection procedures give exact formulas, their GK spectra, and finite-dimensionality criteria; we also describe the dependence on $α$ in the last case. In type I, the GK dimension equals that of the simple even highest weight module for the distinguished Borel. The computations use the established ordinary Lie algebra formulas involving Lusztig's $\mathbf a$-function and Robinson--Schensted insertion.

math.RT↗
arXiv2026

Hopf $2$-cocycles for certain affine algebraic reductive groups

Let $G$ be a reductive affine algebraic group over $\mathbb C$ with toral identity component $T$ and finite component group $K$. We describe finite indecomposable semisimple module categories over $\Rep(G)$ using equivariant data that incorporate the extension class. We prove that every fiber functor is classical and determine the image of its Hopf $2$-cocycle support in $K$ and its intersection with $T$. This yields a minimality criterion involving a finite stabilizer cocycle and an alternating bicharacter on a sublattice of $\widehat T$ of index at most $|K|$. When $\dim T=1$, every Hopf $2$-cocycle for $G$ has finite support of order at most $|K|^2$. We compute all gauge classes for the split and nonsplit extensions with component group $\mathbb{Z}/2\mathbb{Z}$ and inversion action, and compare the support formulas with the Künneth coordinates for commutative direct products. Finally, for an arbitrary reductive group $G$, a minimal Hopf $2$-cocycle in a regular deformation of the trivial Hopf $2$-cocycle forces $G$ to be a torus. This uses locality and rigidity of algebraic matrix coefficients on the semisimple quotient of $G$.

math.RT↗