Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.06885

The Dimension-Shift Category and Its Mellin-Gamma Representation

Andreu Ballus Santacana
View original source ↗Read source document ↗Export metadata ↓

Abstract

We define a thin category $\mathrm{Dim}^+$ of dimension shifts and a category $\mathrm{RadMeas}$ of positive Radon measures with Radon--Nikodym density morphisms. We classify scaling-covariant functors $\mathrm{Dim}^+\to\mathrm{RadMeas}$ whose morphisms are given by homogeneous densities. Gaussian normalization selects a unique functor with values $ d\mu_x(u)=\frac{\pi^{x/2}}{\Gamma(x/2)}u^{x/2-1}\,du. $ Its morphism component yields the radial-integration transport $ R(x,r)=\frac{\pi^r\Gamma(x/2)}{\Gamma(x/2+r)}, $ while the unit-interval observable recovers the Euclidean ball-volume formula $ V(x)=\frac{\pi^{x/2}}{\Gamma(x/2+1)}. $ The two transports differ by the multiplicative coboundary of $\beta(x)=x$, identified with the categorical dimension of the standard object in Deligne's interpolation category $\mathrm{Rep}(O_t)$.

Explore related subjects

math.RTmath.CAmath.CT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Andreu Ballus Santacana. 2025-06-07. The Dimension-Shift Category and Its Mellin-Gamma Representation. https://arxiv.org/abs/2506.06885

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

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Whittaker supermodules over Lie superalgebras

In this paper, we develop a general theory of quasi-Whittaker supermodules over Lie superalgebras induced from an arbitrary ideal. We determine the quasi-Whittaker vectors in universal supermodules, establish an irreducibility criterion, and classify several families of irreducible supermodules. The odd part produces a new irreducibility phenomenon absent from the Lie algebra setting. As applications, we determine all irreducible quasi-Whittaker supermodules over the $N=1$ super Schr\"odinger algebra and the $N=1$ $\frac{3}{2}$-conformal Galilei superalgebra, and over the complete spectrum-generating superalgebra in a special case.

math.RT↗
arXiv2026

Rankin--Selberg integrals of opposite conductor--one newforms

Let $F$ be a nonarchimedean local field of characteristic zero and let $n\geq2$. For $r=n,n+1$, let $\Pi_r$ be an irreducible tempered representation of ${\rm GL}_r(F)$ of conductor one and with trivial central character. We evaluate the Rankin--Selberg integral of opposite newforms in $\Pi_{n+1}\times \Pi_n$ explicitly and show that its central value is nonzero. As an application, this implies a case of Disegni--Zhang's conjecture on the nonvanishing of local relative characters.

math.RT↗
arXiv2026

Obstructions to Jacobi-Finiteness of Quivers with Potentials

We show that Jacobi-finite potentials need not exist on finite $2$-acyclic quivers. Our main tool is a matrix-valued Golod--Shafarevich--Vinberg inequality for quotients of completed path algebras by finitely many, possibly nonhomogeneous, topological relations. Applied to cyclic derivatives, it yields a potential-dependent obstruction to the finite-dimensionality of completed Jacobian algebras. We then construct a purely quiver-level criterion excluding every Jacobi-finite potential on a given quiver, and exhibit a family of quivers for which every potential has an infinite-dimensional Jacobian algebra.

math.RT↗