# Algebra & Combinatorics Seminar

The Algebra & Combinatorics Seminar meets on Fridays from 2:30 to 3:30 pm, in Lecture Hall LH-1 of the IISc Mathematics Department. The organizers are Apoorva Khare and R. Venkatesh.

### Autumn 2018

 Apoorva Khare (IISc Mathematics) Sep 7, 2018 The Diamond Lemma in ring theory

Abstract. I will give a gentle introduction to the Diamond Lemma. This is a useful technique to prove that certain "PBW-type" bases exist of algebras given by generators and relations. In particular, we will see the PBW theorem for usual Lie algebras.

 R. Venkatesh (IISc Mathematics) Sep 14, 2018 Graphs vs. Lie algebras

Abstract. Let $G$ be a finite simple graph. The Lie algebra $\mathfrak{g}$ of $G$ is defined as follows: $\mathfrak{g}$ is generated by the vertices of $G$ modulo the relations $[u, v]=0$ if there is no edge between the vertices $u$ and $v$. Many properties of $\mathfrak{g}$ can be obtained from the properties of $G$ and vice versa. The Lie algebra $\mathfrak{g}$ of $G$ is naturally graded and the graded dimensions of the Lie algebra $\mathfrak{g}$ of $G$ have some deep connections with the vertex colorings of $G$. In this talk, I will explain how to get the generalized chromatic polynomials of $G$ in terms of graded dimensions of the Lie algebra of $G$. We will use this connection to give a Lie theoretic proof of of Stanley's reciprocity theorem of chromatic polynomials.

 No seminar (Muharram) Sep 21, 2018

 Amritanshu Prasad (IMSc, Chennai) Sep 28, 2018 The words that describe symmetric polynomials (speaking in / subsumed by the Eigenfunctions Seminar)