CanaDAM 2021
En ligne, 25 - 28 mai 2021

Polynomials and groups

LOGAN CREW, University of Waterloo
Identities of the Chromatic and Tutte Symmetric Functions  [PDF]

For a graph $G$, its Tutte symmetric function $XB_G$ generalizes both the Tutte polynomial $T_G$ and the chromatic symmetric function $X_G$. We may consider $XB$ as a map from the $t$-extended Hopf algebra $\mathbb{G}[t]$ of labelled graphs to symmetric functions.

In this talk, I show that the kernel of $XB$ is generated by vertex-relabellings and a finite set of modular relations, in the same style as a recent analogous result by Penaguiao for the chromatic symmetric function $X$. Additionally, I give a structural characterization of all local modular relations of the chromatic and Tutte symmetric functions, and prove that there is no single local modification that preserves either function on simple graphs.

This is joint work with Sophie Spirkl.

HOSSEIN TEIMOORI FAAL, Allameh Tabataba'i University, Tehran, Iran
The Generalized Face Handshaking Lemma and Higher Derivatives of Face Polynomials  [PDF]

A face polynomial of a simplicial complex $ \mathcal{K} $ denoted by $ f(\mathcal{K}, x) $ is defined as the generating function of the number of $ k $ - faces in $ \mathcal{K} $ . In this talk, we define the value of a face $ \sigma $ denoted by $ val_{\mathcal{K}}(\sigma) $ as the number of vertices of the link of $ \sigma $ .

A generalization of handshaking lemma for complexes is the following identity: \begin{equation*} \sum_{\sigma:~\sigma~is~a~k-face} val_{\mathcal{K}}(\sigma) = {k+1 \choose 1} f_{k}(\mathcal{K}), ~~~(k \geq 1). \end{equation*} This identitiy implies the well-known combinatorial interpretation of the first derivative of face polynomials. \Our main goal here is to give a generalization of the combinatorial interpretation of the first derivative of face polynomials to higher-order derivatives. We also present some interesting research questions related to face polynomials and their higher derivatives.

AJAY KUMAR, Indian Institute of Technology (BHU), Varanasi, India
Vertex connectivity of superpower graphs of dicyclic groups $T_{4n}$  [PDF]

For a finite group $G,$ the superpower graph $S(G)$ of $G$ is an undirected simple graph with vertex set $G$ and two vertices in $G$ are adjacent in $S(G)$ if and only if the order of one divides the order of the other in the group $G$. Aim of this talk is to provide the tight bounds for the vertex connectivity $\kappa{(S(T_{4n}))}$ of superpower graph $S(T_{4n})$ of dicyclic group $T_{4n}.$

A combinatorial Schur expansion of triangle-free horizontal-strip LLT polynomials  [PDF]

In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial $G_{\lambda}(x;q)$ in some special cases. We associate a weighted graph $\Pi$ to $\lambda$ and we use it to express a linear relation among LLT polynomials. We apply this relation to prove an explicit combinatorial Schur-positive expansion of $G_{\lambda}(x;q)$ whenever $\Pi$ is triangle-free. We also prove that the largest power of $q$ in the LLT polynomial is the total edge weight of our graph.

LAURENCE WIJAYA, Institut Teknologi Bandung
A Relationship Between Cayley-Dickson Process and The Generalized Study Determinant  [PDF]

Study determinant is known as one of replacements for the determinant of matrices with entries in a noncommutative ring. In this work, a generalization of Study determinant is given and show its relationship with the Cayley-Dickson process. Some properties of a non-associative ring obtained by the Cayley- Dickson process with a not necessarily commutative, but associative ring as the initial ring also will be given.