Kalina Mincheva

Algebra and Combinatorics Seminar

We meet weekly on Wednesday at 3:00 PM (Central time), Dinwiddie 102

Organizers: Mahir Can, Tài Huy Hà, Kalina Mincheva.

Schedule: Spring 2022

• Mahir Can, Tulane University.
  The nilpotent variety of an asymptotic semigroup. abstract.
  The asymptotic semigroup of a semisimple group is the semigroup that is obtained by the procedure called "contraction". In this talk, we will discuss the geometry and combinatorics of the nilpotent variety of an asymptotic semigroup. In particular, we will show that the top dimensional homology of this variety affords a permutation representation of the Weyl group.

• Zach Walsh, LSU.
  New lift matroids for group-labeled graphs. abstract.
  Given a graph $G$ with edges labeled by a finite group, a construction of Zaslavsky gives a rank-1 lift of the graphic matroid $M(G)$ which respects the group-labeling. For which groups can we construct a rank-t lift of $M(G)$ with $t > 1$ which respects the group-labeling? We show that this is possible only if the group has a non-trivial partition, and further conjecture that it is possible only if the group is the additive subgroup of a non-prime finite field.

• Tewodros Amdeberhan, Tulane University.
  Critical values of complex polynomials. abstract.
  Given a list of complex numbers, say $w_1,\dots,w_n$ does there exist a complex polynomial $P(z)$ and another list $z_1,\dots,z_n$ so that $P’(z_k)=0$ and $P(z_k)=w_k$ for $k=1,2\dots,n$? We will consider how some basic Analysis and basic Linear Algebra come to the rescue, working in tandem.

• Aram Bingham, UNAM.
  Kronecker coefficients, polytopes, and complexity. abstract.
  The Kronecker coefficients problem is one of the last major open questions in the classical representation theory of symmetric groups. It asks for a combinatorial rule describing the decomposition of tensor products of irreducible symmetric group representations, which is unknown outside of certain special cases. Kronecker coefficients have also been the subject of much recent research motivated by the geometric complexity theory (GCT) program, which hypothesizes efficient computation of these numbers as part of a strategy to separate the computational complexity classes P and NP. We will give an overview of this strategy, explain how the Kronecker problem arises in the context of GCT, and report some progress on computing these coefficients as discrete volumes of polytopes, joint with Ernesto Vallejo.

• Selvi Kara, University of Utah.
  Multi-Rees Algebras of Strongly Stable Ideals. abstract.
  In this talk, we will focus on Rees and multi-Rees algebras of strongly stable ideals. In particular, we will discuss the Koszulness of these algebras through a systematic study of these objects via three parameters: the number of ideals, the number of Borel generators of each ideal, and the degrees of Borel generators. In addition, we utilize combinatorial objects such as fiber graphs to detect Gröbner bases and Koszulness of these algebras.

• Karl H. Hofmann, Tulane University/Darmstadt University.
  This talk will be at 2pm, in room GI-400D.
  Weakly Complete Universal Enveloping Algebras of Profinite-Dimensional Lie Algebras. abstract.
  Every associative algebra $A$ becomes a Lie algebra $A_{\rm Lie}$ with the Lie bracket $[a,b]=ab-ba$. The universal enveloping algebra $U(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ is an associative algebra with identity such that $U(\mathfrak{g})_{\rm Lie}$ contains the Lie subalgebra $\mathfrak{g}$ which generates $U(\mathfrak{g})$ as associative algebra such that every representation $\mathfrak{g}\to A_{\rm Lie}$ for an associative algebra $A$ extends to an algebra homomorphism $U(\mathfrak{g})\to A$. The so called Poincaré-Birkhoff-Witt Theorem secures its existence and structure. In a seminar lecture on September 26 last year I introduced the class of weakly complete topological vector spaces and weakly complete associative unital topological algebras and used the latter to introduce the weakly complete group algebra $\mathbb{K}[G]$ of a topological group $G$ over $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$. In this lecture I shall recall the background for now establishing the theory of a weakly complete universal enveloping algebras $\mathbb{U}(\mathfrak{g})$ of a weakly complete topological Lie algebra $\mathfrak{g}$ and the appropriate Poincaré-Birkhoff-Witt theory. [Joint work with Linus Kramer, University of Münster.]

• Fabrizio Zanello, Michigan Technological University.
  Proof of the Gorenstein Interval Conjecture in low socle degree. abstract.
  In 2009, we proposed the so-called ``Interval Conjectures'' for artinian level algebras, which, if true, would imply a strong (and very natural) structural property for their Hilbert functions. In particular, the Gorenstein Interval Conjecture (GIC) states that if $\alpha \ge 2$, and $(1,\dots,h_i,\dots,h_{e-i},\dots,h_e=1)$ and $(1,\dots,h_i+\alpha,\dots,h_{e-i}+\alpha,\dots,h_e=1)$ are two Gorenstein Hilbert functions that only differ in the symmetric degrees $i$ and $e-i$, then $(1,\dots,h_i+\beta,\dots,h_{e-i}+\beta,\dots,h_e=1)$ is also Gorenstein, for all $\beta =1,\dots, \alpha -1.$ In general, these conjectures are still wide open. In this talk, we outline a proof of the GIC for $e\le 5$. We combine different methods coming from both commutative algebra and algebraic geometry, where suitable ``maximal rank'' properties play a key role. We conclude by discussing some open problems and possible new directions in this area. This is recent joint work with R. Stanley and our former MIT student S.G. Park (J. Algebra, 2019).

• Hop Nguyen, Vietnamese Academy of Science and Technology.
  Binomial expansion of saturated and symbolic powers of sums of ideals. abstract.
  There are two different notions for symbolic powers of ideals existing in the literature, one defined in terms of associated primes, the other in terms of minimal primes. Elaborating on an idea known to Eisenbud, Herzog, Hibi, and Trung, we interpret both notions of symbolic powers as suitable saturations of the ordinary powers. We prove a binomial expansion formula for saturated powers of sums of ideals. This gives a uniform treatment to an array of existing and new results on both notions of symbolic powers of such sums: binomial expansion formulas, computations of depth and regularity, and criteria for the equality of ordinary and symbolic powers. Joint work with H.T. Hà, Ạ.V. Jayanthan, and A. Kumar.

• Hop Nguyen, Vietnamese Academy of Science and Technology.
  Homological invariants of symmetric chains of ideals. abstract.
  Let Inc(N) be the monoid of increasing functions $f:\mathbb{N}\rightarrow \mathbb{N}$ on the positive integers. Let $Sym(N)$ be the direct limit of the symmetric groups on n objects, when $n$ tends to infinity. The monoids $Inc(N)$ and $Sym(N)$ act on the infinite polynomial ring $k[X]=k[x_1,x_2,...]$ (where $k$ is a fixed field) via endomorphisms. Ideals of $k[X]$ which are invariants under the $Inc(N)$- or $Sym(N)$-action tend to have significant finiteness properties, due to their huge individual orbits. We discuss recent results and problems on the homological invariants of $Inc(N)$- and $Sym(N)$-invariant ideals. Joint work with Dinh Van Le.

• Mark Skandera, Lehigh University.
  Symmetric generating functions and permanents of totally nonnegative matrices. abstract.
  For each element $z$ of the symmetric group algebra we define a symmetric generating function $Y(z) = Σ_\lambda \epsilon^\lambda(z) m_\lambda$, where $\epsilon^\lambda$ is the induced sign character indexed by $\lambda$. Expanding $Y(z)$ in other symmetric function bases, we obtain other trace evaluations as coefficients. We show that all symmetric functions in $span_Z\{m_\lambda\}$ are $Y(z)$ for some $z$ in $Q[S_n]$. Using this fact and chromatic symmetric functions, we give new interpretations of permanents of totally nonnegative matrices. For the full paper, see arXiv:2010.00458v2.

• Friedrich Wagemann, Universite de Nantes, France,
  On Lie's Third Theorem for Leibniz Algebras. abstract.
  Leibniz algebras generalize Lie algebras in the sense that the bracket is not necessarily antisymmetric. We will describe in our talk attempts to integrate Leibniz algebras into Lie racks. Here a rack is an algebraic structure more general than the notion of a group; in fact, one retains from the notion of a group only the conjugation map. M. Kinyon showed in 2007 that the tangent space of a Lie rack carries naturally the structure of a Leibniz algebra. S. Covez showed in 2010 that Leibniz algebras integrate into local Lie racks. Many other integration procedures have been proposed since then. We will focus on the integration procedure of Bordemann-Wagemann (2017), where a general Leibniz algebra is integrated into a Lie rack which is an affine bundle over a Lie group such that in case the Leibniz algebra is a Lie algebra, one obtains the standard integration of Lie algebras. The drawback of this procedure is that it is not functorial.