arXiv subjects

Steven V Sam

Publications and source records attributed to Steven V Sam.

At least 19 recordsLinked to original sources

arXiv2026

Cohomology of complex supertori

We consider supertori which are quotients of affine superspace by translations by algebraically independent odd parameters. Specifically, we describe the ring structure of its space of global sections by generators and relations and completely determine the coherent cohomology groups of its structure sheaf by reducing it to a problem of Lie algebra cohomology. We also show that Poincar\'e duality on sheaf cohomology is compatible with that of the group cohomology of the translation group and give explicit tables of examples. Finally, we compute the Picard groups of these supertori.

math.AG↗

arXiv2026

Powers of binary forms and derived Hermite reciprocity

For $a,b \ge 1$, Hilbert found in 1886 a collection of polynomial equations that cut out set-theoretically the variety X parametrizing a-th powers of binary forms of degree b. We determine the ideal of all polynomials vanishing on X, showing that it is generated in degree b+1 and that it has a linear minimal free resolution. We do this by generalizing results of Abdesselam and Chipalkatti on an analogue of the Foulkes--Howe map and by establishing a derived analogue of the classical Hermite reciprocity theorem for complexes of ${\rm SL}_2$-representations. In our investigation, we are led to the ideal generated by the subrepresentation ${\rm Sym}^{ab}({\Bbb C}^2) \subset {\rm Sym}^a({\rm Sym}^b {\Bbb C}^2)$. We determine its Castelnuovo--Mumford regularity in general and the minimal free resolution for small values of b.

math.AC↗

arXiv2025

Kostant $\rho$-decomposition of homology I. Finite-dimensional representations

We give explicit, uniform formulas for the graded characters and total ranks of the Lie algebra homology of finite-dimensional representations in all classical types. In many cases, these compute the Tor groups of finite length modules over polynomial rings, and this is the first in a series of papers to investigate total rank conjectures from this perspective. These formulas refine and generalize the classical $\rho$-decomposition of Kostant, and in particular we prove that the characters involved exhibit three structural phenomena: divisibility (by a large power of 2), equidistribution, and uniform factorization formulas.

math.RT↗

arXiv2025

An infinite-dimensional character coincidence between Lie algebras of type B and BC

We utilize an isomorphism between the character rings of the odd orthogonal group and the orthosymplectic supergroup to understand equivariant positivity properties of the type B quadric hypersurface ring. Our main result establishes a well-behaved functorial construction of Schur modules ``with respect to'' the quadric hypersurface ring, an essential fact used by the authors in previous work to construct pure free resolutions. Our techniques combine ideas from commutative algebra, Lie theory, and algebraic geometry to understand representations of orthosymplectic supergroups and their corresponding type B counterparts.

math.RT↗

arXiv2024

Cohomology of flag supervarieties and resolutions of determinantal ideals. II

We compute the coherent cohomology of the structure sheaf of complex periplectic Grassmannians. In particular, we show that it can be decomposed as a tensor product of the singular cohomology ring of a Grassmannian for either the symplectic or orthogonal group together with a semisimple representation of the periplectic Lie supergroup. The restriction of the latter to its even subgroup has an explicit multiplicity-free description in terms of Schur functors and is closely related to syzygies of (skew-)symmetric determinantal ideals. We develop tools for studying splitting rings for Coxeter groups of types BC and D, which may be of independent interest.

math.AG↗

arXiv2024

From total positivity to pure free resolutions

Using the Jacobi-Trudi identity as a base, we establish parallels between the theory of totally positive integer sequences and Koszul algebras. We then focus on the case of quadric hypersurface rings and use this parallel to construct new analogues of Schur modules. We investigate some of their Lie-theoretic properties (and in more detail in a followup article) and use them to construct pure free resolutions for quadric hypersurface rings which are completely analogous to the construction given by Eisenbud, Fl{\o}ystad, and Weyman in the case of polynomial rings.

math.AC↗

arXiv2024

Borel-Weil factorization for super Grassmannians

This article deals with computing the cohomology of Schur functors applied to tautological bundles on super Grassmannians. We show that in a range of cases, the cohomology is a free module over the cohomology of the structure sheaf and that the space of generators is an irreducible representation of the general linear supergroup that can be constructed via explicit multilinear operations. Our techniques come from commutative algebra: we relate this cohomology calculation to Tor groups of certain algebraic varieties.

math.RT↗

arXiv2022

On the cohomology of tautological bundles over Quot schemes of curves

We consider tautological bundles and their exterior and symmetric powers on the Quot scheme over the projective line. We prove and conjecture several statements regarding the vanishing of their higher cohomology, and we describe their spaces of global sections via tautological constructions. To this end, we make use of the embedding of the Quot scheme as an explicit local complete intersection in the product of two Grassmannians, studied by Str{\o}mme. This allows us to construct resolutions with vanishing cohomology for the tautological bundles and their exterior and symmetric powers. We further illustrate our approach with a few additional cohomological calculations.

math.AG↗

arXiv2022

Polynomial representations of the Witt Lie algebra

The Witt algebra W_n is the Lie algebra of all derivations of the n-variable polynomial ring V_n=C[x_1, ..., x_n] (or of algebraic vector fields on A^n). A representation of W_n is polynomial if it arises as a subquotient of a sum of tensor powers of V_n. Our main theorems assert that finitely generated polynomial representations of W_n are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of Fin^op, where Fin is the category of finite sets. We also show that polynomial representations of W_n are equivalent to polynomial representations of the endomorphism monoid of A^n. These equivalences are a special case of an operadic version of Schur--Weyl duality, which we establish.

math.RT↗

arXiv2022

A note on projective dimension over twisted commutative algebras

Let $M$ be a finitely generated module over a free twisted commutative algebra $A$ that is finitely generated in degree one. We show that the projective dimension of $M({\bf C}^n)$ as an $A({\bf C}^n)$-module is eventually linear as a function of $n$. This confirms a conjecture of Le, Nagel, Nguyen, and R\"omer for a special class of modules.

math.AC↗

arXiv2022

The representation theory of Brauer categories II: curried algebra

A representation of $\mathfrak{gl}(V)=V \otimes V^*$ is a linear map $\mu \colon \mathfrak{gl}(V) \otimes M \to M$ satisfying a certain identity. By currying, giving a linear map $\mu$ is equivalent to giving a linear map $a \colon V \otimes M \to V \otimes M$, and one can translate the condition for $\mu$ to be a representation to a condition on $a$. This alternate formulation does not use the dual of $V$, and makes sense for any object $V$ in a tensor category $\mathcal{C}$. We call such objects representations of the curried general linear algebra on $V$. The currying process can be carried out for many algebras built out of a vector space and its dual, and we examine several cases in detail. We show that many well-known combinatorial categories are equivalent to the curried forms of familiar Lie algebras in the tensor category of linear species; for example, the titular Brauer category "is" the curried form of the symplectic Lie algebra. This perspective puts these categories in a new light, has some technical applications, and suggests new directions to explore.

math.RT↗

arXiv2022

Jacobi-Trudi formulas and determinantal varieties

Gessel gave a determinantal expression for certain sums of Schur functions which visually looks like the classical Jacobi-Trudi formula. We explain the commonality of these formulas using a construction of Zelevinsky involving BGG complexes and use this explanation to generalize this formula in a few different directions.

math.CO↗

arXiv2021

On some modules supported in the Chow variety

The study of Chow varieties of decomposable forms lies at the confluence of algebraic geometry, commutative algebra, representation theory and combinatorics. There are many open questions about homological properties of Chow varieties and interesting classes of modules supported on them. The goal of this note is to survey some fundamental constructions and properties of these objects, and to propose some new directions of research. Our main focus will be on the study of certain maximal Cohen-Macaulay modules of covariants supported on Chow varieties, and on defining equations and syzygies. We also explain how to assemble Tor groups over Veronese subalgebras into modules over a Chow variety, leading to a result on the polynomial growth of these groups.

math.AC↗

arXiv2021

Cohomology of flag supervarieties and resolutions of determinantal ideals

We study the coherent cohomology of generalized flag supervarieties. Our main observation is that these groups are closely related to the free resolutions of (certain generalizations of) determinantal ideals. In the case of super Grassmannians, we completely compute the cohomology of the structure sheaf: it is composed of the singular cohomology of a Grassmannian and the syzygies of a determinantal variety. The majority of the work involves studying the geometry of an analog of the Grothendieck-Springer resolution associated to the super Grassmannian; this takes place in the world of ordinary (non-super) algebraic geometry. Our work gives a conceptual explanation of the result of Pragacz-Weyman that the syzygies of determinantal ideals admit an action of the general linear supergroup. In a subsequent paper, we will treat other flag supervarieties in detail.

math.AG↗

arXiv2021

A Bruhat atlas for the Mehta-van der Kallen stratification of $T^* GL_n/B$

Mehta and van der Kallen put a Frobenius splitting on the type A cotangent bundle $T^* GL_n/B$, thereby defining a stratification by compatibly split subvarieties, and they determined a few of the elements of this stratification. We embed $T^* GL_n/B$ as a stratum in a larger stratified (and Frobenius split) space $GL_n/B \times Mat_n$ whose stratification we determine, thereby giving a full description of the one of Mehta-van der Kallen. The main technique is to endow $GL_n/B \times Mat_n$ with a Bruhat atlas, covering it with open sets that are stratified-isomorphic to Bruhat cells (in $GL_{2n}/B_{2n}$). Among the consequences are that each stratum closure is normal and Cohen-Macaulay.

math.AG↗

arXiv2021

Hermite reciprocity and Schwarzenberger bundles

Hermite reciprocity refers to a series of natural isomorphisms involving compositions of symmetric, exterior, and divided powers of the standard $SL_2$-representation. We survey several equivalent constructions of these isomorphisms, as well as their recent applications to Green's Conjecture on syzygies of canonical curves. The most geometric approach to Hermite reciprocity is based on an idea of Voisin to realize certain multilinear constructions cohomologically by working on a Hilbert scheme of points. We explain how in the case of ${\bf P}^1$ this can be reformulated in terms of cohomological properties of Schwarzenberger bundles. We then proceed to study these bundles from several perspectives: We show that their exterior powers have supernatural cohomology, arising as special cases of a construction of Eisenbud and Schreyer. We recover basic properties of secant varieties $\Sigma$ of rational normal curves (normality, Cohen-Macaulayness, rational singularities) by considering their desingularizations via Schwarzenberger bundles, and applying the Kempf-Weyman geometric technique. We show that Hermite reciprocity is equivalent to the self-duality of the unique rank one Ulrich module on the affine cone $\widehat{\Sigma}$ of some secant variety, and we explain how for a Schwarzenberger bundle of rank $k$ and degree $d\ge k$, Hermite reciprocity can be viewed as the unique (up to scaling) non-zero section of $(Sym^k\mathcal{E})(-d+k-1)$.

math.AG↗

arXiv2021

On the geometry and representation theory of isomeric matrices

The space of $n \times m$ complex matrices can be regarded as an algebraic variety on which the group ${\bf GL}_n \times {\bf GL}_m$ acts. There is a rich interaction between geometry and representation theory in this example. In an important paper, de Concini, Eisenbud, and Procesi classified the equivariant ideals in the coordinate ring. More recently, we proved a noetherian result for families of equivariant modules as $n$ and $m$ vary. In this paper, we establish analogs of these results for the space of $(n|n) \times (m|m)$ isomeric matrices with respect to the action of ${\bf Q}_n \times {\bf Q}_m$, where ${\bf Q}_n$ is the automorphism group of the isomeric structure (commonly known as the "queer supergroup"). Our work is motivated by connections to the Brauer category and the theory of twisted commutative algebras.

math.RT↗

arXiv2020

Supersymmetric monoidal categories

We develop the idea of a supersymmetric monoidal supercategory, following ideas of Kapranov. Roughly, this is a monoidal category in which the objects and morphisms are ${\bf Z}/2$-graded, equipped with isomorphisms $X \otimes Y \to Y \otimes X$ of parity $\vert X \vert \vert Y \vert$ on homogeneous objects. There are two fundamental examples: the groupoid of spin-sets, and the category of queer vector spaces equipped with the half tensor product; other important examples can be derived from these (such as the category of linear spin species). There are also two general constructions. The first is the exterior algebra of a supercategory (due to Ganter--Kapranov). The second is a construction we introduce called Clifford eversion. This defines an equivalence between a certain 2-category of supersymmetric monoidal supercategories and a corresponding 2-category of symmetric monoidal supercategories. We use our theory to better understand some aspects of the queer superalgebra, such as certain factors of $\sqrt{2}$ in the theory of Q-symmetric functions and Schur--Sergeev duality.

math.CT↗