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Špela Špenko

Publications and source records attributed to Špela Špenko.

32 records · Page 2Linked to original sources
arXiv2018

Positive trace polynomials and the universal Procesi-Schacher conjecture

Positivstellensatz is a fundamental result in real algebraic geometry providing algebraic certificates for positivity of polynomials on semialgebraic sets. In this article Positivstellensätze for trace polynomials positive on semialgebraic sets of $n\times n$ matrices are provided. A Krivine-Stengle-type Positivstellensatz is proved characterizing trace polynomials nonnegative on a general semialgebraic set $K$ using weighted sums of hermitian squares with denominators. The weights in these certificates are obtained from generators of $K$ and traces of hermitian squares. For compact semialgebraic sets $K$ Schmüdgen- and Putinar-type Positivstellensätze are obtained: every trace polynomial positive on $K$ has a sum of hermitian squares decomposition with weights and without denominators. The methods employed are inspired by invariant theory, classical real algebraic geometry and functional analysis. Procesi and Schacher in 1976 developed a theory of orderings and positivity on central simple algebras with involution and posed a Hilbert's 17th problem for a universal central simple algebra of degree $n$: is every totally positive element a sum of hermitian squares? They gave an affirmative answer for $n=2$. In this paper a negative answer for $n=3$ is presented. Consequently, including traces of hermitian squares as weights in the Positivstellensätze is indispensable.

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arXiv2018

On the noncommutative Bondal-Orlov conjecture for some toric varieties

We show that all toric noncommutative crepant resolutions (NCCRs) of affine GIT quotients of "weakly symmetric" unimodular torus representations are derived equivalent. This yields evidence for a non-commutative extension of a well known conjecture by Bondal and Orlov stating that all crepant resolutions of a Gorenstein singularity are derived equivalent. We prove our result by showing that all toric NCCRs of the affine GIT quotient are derived equivalent to a fixed Deligne-Mumford GIT quotient stack associated to a generic character of the torus. This extends a result by Halpern-Leistner and Sam which showed that such GIT quotient stacks are a geometric incarnation of a family of specific toric NCCRs constructed earlier by the authors.

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arXiv2018

Semi-orthogonal decompositions of GIT quotient stacks

If G is a reductive group which acts on a linearized smooth scheme $X$ then we show that under suitable standard conditions the derived category of coherent sheaves of the corresponding GIT quotient stack $X^{ss}/G$ has a semi-orthogonal decomposition consisting of derived categories of coherent sheaves of rings on the categorical quotient $X^{ss}/\!/G$ which are locally of finite global dimension. One of the components of the decomposition is a certain non-commutative resolution of $X^{ss}/\!/G$ constructed earlier by the authors. As a concrete example we obtain in the case of odd Pfaffians a semi-orthogonal decomposition of the corresponding quotient stack in which all the parts are certain specific non-commutative crepant resolutions of Pfaffians of lower or equal rank which had also been constructed earlier by the authors. In particular this semi-orthogonal decomposition cannot be refined further since its parts are Calabi-Yau. The results in this paper also complement a result by Halpern-Leistner (and similar results by Ballard-Favero-Katzarkov and Donovan-Segal) that asserts the existence of a semi-orthogonal decomposition of the derived category of $X/G$ in which one of the components is the derived category of $X^{ss}/G$.

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arXiv2017

The Frobenius morphism in invariant theory

Let $R$ be the homogeneous coordinate ring of the Grassmannian $\mathbb{G}=\operatorname{Gr}(2,n)$ defined over an algebraically closed field of characteristic $p>0$. In this paper we give a completely characteristic free description of the decomposition of $R$, considered as a graded $R^p$-module, into indecomposables ("Frobenius summands"). As a corollary we obtain a similar decomposition for the Frobenius pushforward of the structure sheaf of $\mathbb{G}$ and we obtain in particular that this pushforward is almost never a tilting bundle. On the other hand we show that $R$ provides a "noncommutative resolution" for $R^p$ when $p\ge n-2$, generalizing a result known to be true for toric varieties. In both the invariant theory and the geometric setting we observe that if the characteristic is not too small the Frobenius summands do not depend on the characteristic in a suitable sense. In the geometric setting this is an explicit version of a general result by Bezrukavnikov and Mirković on Frobenius decompositions for partial flag varieities. We are hopeful that it is an instance of a more general "$p$-uniformity" principle.

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arXiv2017

Generalised Witt algebras and idealizers

Let $\Bbbk$ be an algebraically closed field of characteristic zero, and let $Γ$ be an additive subgroup of $\Bbbk$. Results of Kaplansky-Santharoubane and Su classify intermediate series representations of the generalised Witt algebra $W_Γ$ in terms of three families, one parameterised by ${\mathbb A}^2$ and two by ${\mathbb P}^1$. In this note, we use the first family to construct a homomorphism $Φ$ from the enveloping algebra $U(W_Γ)$ to a skew extension of ${\Bbbk}[a,b]$. We show that the image of $Φ$ is contained in a (double) idealizer subring of this skew extension and that the representation theory of idealizers explains the three families. We further show that the image of $U(W_Γ)$ under $Φ$ is not left or right noetherian, giving a new proof that $U(W_Γ)$ is not noetherian. We construct $Φ$ as an application of a general technique to create ring homomorphisms from shift-invariant families of modules. Let $G$ be an arbitrary group and let $A$ be a $G$-graded ring. A graded $A$-module $M$ is an intermediate series module if $M_g$ is one-dimensional for all $g \in G$. Given a shift-invariant family of intermediate series $A$-modules parametrised by a scheme $X$, we construct a homomorphism $Φ$ from $A$ to a skew-extension of ${\Bbbk}[X]$. The kernel of $Φ$ consists of those elements which annihilate all modules in $X$.

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arXiv2017

Non-commutative resolutions of quotient singularities

In this paper we generalize standard results about non-commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups. We show in particular that quotient singularities for reductive groups always have non-commutative resolutions in an appropriate sense. Moreover we exhibit a large class of such singularities which have (twisted) non-commutative crepant resolutions. We discuss a number of examples, both new and old, that can be treated using our methods. Notably we prove that twisted non-commutative crepant resolutions exist in previously unknown cases for determinantal varieties of symmetric and skew-symmetric matrices. In contrast to almost all prior results in this area our techniques are algebraic and do not depend on knowing a commutative resolution of the singularity.

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arXiv2015

Sweeping words and the length of a generic vector subspace of M_n(F)

The main result of this short note is a generic version of Paz's conjecture on the lengths of generating sets in matrix algebras. Consider a generic g-tuple A=(A_1,..., A_g) of nxn matrices over a field. We show that whenever $g^{2d}\geq n^2$, the set of all words of degree 2d in A spans the full nxn matrix algebra. Our proofs use generic matrices, are combinatorial and depend on the construction of a special kind of directed multigraphs with few edge-disjoint walks.

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arXiv2014

Functional identities on matrix algebras

Complete solutions of functional identities $\sum_{k\in K}F_k(\bar{x}_m^k)x_k = \sum_{l\in L}x_lG_l(\bar{x}_m^l)$ on the matrix algebra $M_n(\mathbb{F})$ are given. The nonstandard parts of these solutions turn out to follow from the Cayley-Hamilton identity.

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arXiv2014

Free function theory through matrix invariants

In this article we introduce powerful tools and techniques from invariant theory to free analysis. This enables us to study free maps with involution. These maps are free noncommutative analogs of real analytic functions of several variables. With examples we demonstrate that they do not exhibit strong rigidity properties of their involution-free free counterparts. We present a characterization of polynomial free maps via properties of their finite-dimensional slices. This is used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are given by series of generalized polynomials for which we obtain an invariant-theoretic characterization. Finally, we give an inverse and implicit function theorem for free maps with involution.

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arXiv2014

Quasi-identities on matrices and the Cayley-Hamilton polynomial

We consider certain functional identities on the matrix algebra $M_n$ that are defined similarly as the trace identities, except that the "coefficients" are arbitrary polynomials, not necessarily those expressible by the traces. The main issue is the question of whether such an identity is a consequence of the Cayley-Hamilton identity. We show that the answer is affirmative in several special cases, and, moreover, for every such an identity $P$ and every central polynomial $c$ with zero constant term there exists $m\in\mathbb{N}$ such that the affirmative answer holds for $c^mP$. In general, however, the answer is negative. We prove that there exist antisymmetric identities that do not follow from the Cayley-Hamilton identity, and give a complete description of a certain family of such identities.

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arXiv2013

A tracial Nullstellensatz

The main result of this note is a tracial Nullstellensatz for free noncommutative polynomials evaluated at tuples of matrices of all sizes: Suppose f_1,...,f_r,f are free polynomials, and tr(f) vanishes whenever all tr(f_j) vanish. Then either 1 or f is a linear combination of the f_j modulo sums of commutators.

math.RA↗
arXiv2013

Functional identities of one variable

Let $A$ be a centrally closed prime algebra over a characteristic 0 field $k$, and let $q:A\to A$ be the trace of a $d$-linear map (i.e., $q(x)=M(x,...,x)$ where $M:A^d\to A$ is a $d$-linear map). If $[q(x),x]=0$ for every $x\in A$, then $q$ is of the form $q(x) =\sum_{i=0}^{d} μ_i(x)x^i$ where each $μ_i$ is the trace of a $(d-i)$-linear map from $A$ into $k$. For infinite dimensional algebras and algebras of dimension $>d^2$ this was proved by Lee, Lin, Wang, and Wong in 1997. In this paper we cover the remaining case where the dimension is $ \le d^2$. Using this result we are able to handle general functional identities of one variable on $A$; more specifically, we describe the traces of $d$-linear maps $q_i:A\to A$ that satisfy $\sum_{i=0}^m x^i q_i(x)x^{m-i}\in k$ for every $x\in A$.

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arXiv2013

On the image of a noncommutative polynomial

Let $F$ be an algebraically closed field of characteristic zero. We consider the question which subsets of $M_n(F)$ can be images of noncommutative polynomials. We prove that a noncommutative polynomial $f$ has only finitely many similarity orbits modulo nonzero scalar multiplication in its image if and only if $f$ is power-central. The union of the zero matrix and a standard open set closed under conjugation by $GL_n(F)$ and nonzero scalar multiplication is shown to be the image of a noncommutative polynomial. We investigate the density of the images with respect to the Zariski topology. We also answer Lvov's conjecture for multilinear Lie polynomials of degree at most 4 affirmatively.

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