#### Alladi Ramakrishnan Hall

#### Bases for root spaces of Borcherds-Kac-Moody algebras

#### R. Venkatesh

##### IIT Madras

*Let $G$ be the BKM algebra. We consider the roots of $G$ of the form $\sum_{i\in I} k_i\alpha_i$ with the $k_i\leq 1$ for real simple roots $\alpha_i$. We prove that the root multiplicities of these roots have a close relationship with the (multi-)chromatic polynomial of the graph of $G$. Using this relationship, we construct bases of the root spaces of $G$ associated to these roots. This is joint work with G. Arunkumar and Deniz Kus.*

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