# Pfaffians, Tensors, and Applications

## Abstracts for the talks

**Hirotachi Abo ***University of Idaho***Compatibility of eigenconfigurations**

The left eigenvectors and right eigenvectors of a square matrix are distinct, but they are compatible. The purpose of this talk is two-fold; the first is to extend the concepts of left eigenvectors and right eigenvectors of a matrix to tensors, and the second is to explore the compatibility of such concepts for a binary tensor.

**Daniele Agostini ***Universitaet Tuebingen***Sato Grassmannian and polynomial theta functions**

The Sato Grassmannian is a certain infinite dimensional version of

the Grassmannian, which encodes all solutions to the KP hierarchy, an infinite series of partial differential equations. On the other hand, such solutions can be constructed explicitly via the theta function of an algebraic curve. I will show that when the curve is particularly degenerate, we get rational solutions to the KP equation, and I will use the Sato Grassmannian as an essential tool.

This is joint work with Turku Celik and John Little.

**Vincenzo Antonelli ***Politecnico di Torino***Steiner representations of hypersurfaces**

Let X ⊆ ℙ^{n+1} be an integral hypersurface of degree d. The description of hypersurfaces in ℙ^{n+1} as zero loci of suitable square matrices (possibly with some further properties, e.g. with linear entries, symmetric, skew–symmetric, etc.) is a very classical topic in algebraic geometry. In this talk we show that each locally Cohen–Macaulay instanton sheaf E on X with respect to _{X} ⊗_{ℙn+1}(1) yields the existence of Steiner bundles and F on ℙ^{n+1} of the same rank r and a morphism φ: G(−1)^{∨} such that the form deﬁning X to the power (E) is exactly det(φ). In particular, we show that the form deﬁning a smooth integral surface in ℙ^{3} is the pfaﬃan of some skew–symmetric morphism φ: F(−1) → F^{∨}, where F is a suitable Steiner bundle on ℙ^{3} of suﬃciently large even rank. Finally we deal with the case of cubic fourfolds in ℙ^{5}, showing how the existence of Steiner pfaﬃan representations is related to the existence of particular subvarieties of the cubic. This is a joint work with Gianfranco Casnati.

**Ada Boralevi ***Politecnico di Torino***Spaces of matrices with rank conditions**

I will give an overview on linear spaces of matrices with bounded and constant rank, in connection with the study of vector bundles on projective varieties. After a general introduction, I will focus on the skew-symmetric case, where one deals with linear spaces inside the Pfaffian hypersurface. If time permits, I will also give some results and open problems in the quadratic case.

**Willem de Graaf ***University of Trento***Classifications of complex and real tensors**

**Sara Angela Filippini ***Jagiellonian University***Free resolutions from opposite Schubert varieties in minuscule homogeneous spaces**

Free resolutions F_{∙} of Cohen-Macaulay and Gorenstein ideals have been investigated for a long time. An important task is to determine generic resolutions for a given format rkF_{i}. Starting from the Kac-Moody Lie algebra associated to a T-shaped graph T˙p,q,r, Weyman constructed generic rings for every format of resolutions of length 3. When the graph T_{p,q,r} is Dynkin, these generic rings are Noetherian. Sam and Weyman showed that for all Dynkin types the ideals of the intersections of certain Schubert varieties of codimension 3 with the opposite big cell of the homogeneous spaces G(T_{p,q,r})∕P, where P is a speciﬁed maximal parabolic subgroup, have resolutions of the given format. In joint work with J. Torres and J. Weyman we study the case of Schubert varieties in minuscule homogeneous spaces and ﬁnd resolutions of some well-known Cohen-Macaulay and Gorenstein ideals of higher codimension.

**Dragos Fratila ***Universite de Strasbourg***Jordan--Chevalley decomposition for G-bundles on genus one curves**

(joint with Sam Gunningham and Penghui Li)

The Jordan--Chevalley decomposition expresses a matrix as a sum of a diagonalizable and a nilpotent matrix that commute. In fact such a decomposition holds both for the Lie algebra and for the group and more generally for any reductive group. One uses extensively this result in representation theory, in the study of the conjugacy classes of matrices and their geometry, etc. In this talk I will explain that there is another situation when one can decompose a "G"-object like this, namely that of semistable G-bundles on elliptic curves: it says that any G-bundle can be written essentialy uniquely as a product of a unipotent bundle and a semisimple bundle. Since G-bundles do not admit a multiplication/addition as is the case for Lie algebras or Lie groups, we need first to make sense of what such a decomposition should mean.

I will present various ways of thinking about the Jordan--Chevalley decomposition and focus on the (geometric) one that admits a generalization to G-bundles. Moreover we will see that degenerating the elliptic curve to a nodal or a cusp leads us back to the conjugacy classes in the Lie group, resp. Lie algebra, and to the corresponding Jordan--Chevalley decomposition, exhibiting thus the trichotomy "rational, trigonometric, elliptic".

Using a tannakian description of semistable G-bundles one can also show that, for non supersingular elliptic curves, the elliptic unipotent cone is isomorphic to the unipotent/nilpotent cone in the group/Lie algebra.

**Claire Gilson ***University of Glasgow***Pfaffian Solutions to Non-Commutative Integrable Systems**

**Lukas Kühne ***Universität Bielefeld***Matroids and Algebra**

A matroid is a combinatorial object based on an abstraction of linear independence in vector spaces and forests in graphs. I will discuss how matroid theory interacts with algebra via the so-called von Staudt constructions. These are combinatorial gadgets to encode polynomials in matroids.

I will discuss generalized matroid representations as arrangements over division rings, subspace arrangements, and probability space representations together with their relation to group theory.

As an application, this yields a proof that the conditional independence implication problem from information theory is undecidable.

Based on joint work with Rudi Pendavingh and Geva Yashfe.

**Daniel Plaumann ***Technische Universität Dortmund***Determinantal hypersurfaces**

Linear determinantal representations of polynomials have been studied in real and convex algebraic geometry ever since Helton and Vinnikov established the connection with the study of linear matrix inequalities and hyperbolic polynomials. In this talk, I will give a brief overview with examples, open questions and current developments.

**Tobias Rossmann ***University of Galway***Tensors and group orbits**

This talk will be devoted to algebraic and geometric structures surrounding the enumeration of orbits of groups. A particular focus will be on linear orbits and conjugacy classes of unipotent groups. In this setting, bilinear maps and other tensor-related structures recently emerged as powerful tools that can help us understand the absence or presence of certain geometric features.

**Pierpaola Santarsiero ***MPI-MiS***Representation theory of signature tensors**

In this talk we will look at signature tensors from the perspective given by representation theory and we will understand the connection between the vector spaces determinded by fixed Lyndon words and Schur modules. Next, we will try to characterize when a signature tensor of some given type is actually a (skew)-symmetric tensor.

**Javier Sendra Arranz ***Max Planck Institute for Mathematics in the Sciences***The Hessian map of hypersurfaces**

Let F be a degree d form in n+1 variables. The Gauss map sends each smooth point of the hypersurface defined by F to the evaluation of the Jacobian of F at the point. The image of the Gauss map is the dual variety of the hypersurface. After applying the Gauss map again, the original hypersurface is recovered. In this talk we study the analogous map for the Hessian matrix. The Hessian map of F is a rational map that sends a point of the hypersurface defined by F to the evaluation of the Hessian matrix of F at the point. We study some properties of this morphism as the birationality and the smoothness. In the case d=3, we introduce the variety of k--planes containing the span of gradients of degree 3 forms. The language of symmetric tensor provides a dictionary between this variety and the Hessian map that allows us to derive an effective method for recovering the initial hypersurface from its image through the Hessian map.

**Tim Seynnaeve ***KU Leuven***Universal equations for Lagrangian and isotropic Grassmannians**

It is known that any ordinary Grassmannian Gr(k,n) in its Plücker embedding is set-theoretically (but not scheme-theoretically) cut out by pulling back the unique defining equation of Gr(2,4) along natural projection and contraction maps. We prove an analogous result for Lagrangian and isotropic Grassmannians. This talk is based on joint work with Nafie Tairi.

**Alexandru Suciu ***Northeastern University***Cohomology jump loci, Poincaré duality, and Pfaffians**

The cohomology jump loci of a space are of several types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems; the resonance varieties, constructed from information encoded in the cohomology ring; and the complements to the Bieri-Neumann-Strebel-Renz invariants, which are defined in terms of Novikov-Sikorav homology. In this talk, I will explore the geometry of these sets and the delicate interplay between them, especially in the context of compact, orientable 3-manifolds, where Poincaré duality and Pfaffians play an important role.

**James Wilson ***Colorado State University***Tensor-Ideal-Operator Correspondence**

Tensor generalizations of standard linear algebra (for example rank, eigen and singular values) are now known to be NP-hard. But tensors are not only generalized linear algebra. Tensors can just as well be studied as generalized non-associative algebra. I will survey a number of polynomial-time algorithm for tensor structure made possible from this perspective leading to a summary result: Lie algebras, not associative rings, are the universal coefficients for tensor products.

**Robert Zeier ***Forschungszentrum Jülich***From algebra and symmetries to quantum computing**

How can better algebraic and Lie-algebraic methods help to more efficiently analyze, design, control, and program large quantum computers? Symmetries will be the key! High-dimensional quantum systems and their dynamics in emerging quantum technologies are a key application of tensors and tensor-product structures. We have developed symmetry methods [J. Math. Phys. 52(11):113510, 2011; J. Math. Phys. 56(8):081702, 2015; Phys. Rev. A 92(4):042309, 2015] for quantum control theory to answer simulability questions for quantum computing devices. In our approach, Lie algebras are characterized using so-called quadratic symmetries related to the tensor power of a representation. Symmetries are computed (and defined) as the linear space of all matrices commuting with a set of Lie-algebra generators using efficient sparse linear algebra in Magma. But we wonder if more efficient approaches might be possible following ideas of Wilson and Maglione. Capabilities of quantum computers are related to identifying the generated Lie algebra (as in the work of de Graaf), preferably without constructing it explicitly. Algebraic methods such as the Meataxe algorithm can help to identify Lie algebras from symmetries. We close by outlining recent developments which include the use of the Weisfeiler-Leman algorithm to relate graph properties to symmetries as well as a Lie-algebra classification related to so-called variational quantum algorithms.

## Date and Location

**November 28 - 30, 2022**

Max Planck Institute for Mathematics in the Sciences

E1 05 (Leibniz-Saal)

Inselstr. 22

04103 Leipzig

## Scientific Organizers

**Daniele Faenzi**

Université de Bourgogne, CNRS**Joshua Maglione**

Otto-von-Guericke-Universität**Mima Stanojkovski**

Università di Trento

## Administrative Contact

**Saskia Gutzschebauch**

MPI for Mathematics in the Sciences

Contact by Email

**Mirke Olschewski**

MPI for Mathematics in the Sciences

Contact by Email