## View abstract

### Session S16 - Quantum symmetries

Wednesday, July 14, 12:10 ~ 12:35 UTC-3

## Markov chains from Weyl modules for quantum $sl_2$

### Georgia Benkart

#### University of Wisconsin-Madison, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak0d754b361dd79d6c13e73dba0a96257b').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy0d754b361dd79d6c13e73dba0a96257b = 'b&#101;nk&#97;rt' + '&#64;'; addy0d754b361dd79d6c13e73dba0a96257b = addy0d754b361dd79d6c13e73dba0a96257b + 'm&#97;th' + '&#46;' + 'w&#105;sc' + '&#46;' + '&#101;d&#117;'; var addy_text0d754b361dd79d6c13e73dba0a96257b = 'b&#101;nk&#97;rt' + '&#64;' + 'm&#97;th' + '&#46;' + 'w&#105;sc' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak0d754b361dd79d6c13e73dba0a96257b').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy0d754b361dd79d6c13e73dba0a96257b + '\'>'+addy_text0d754b361dd79d6c13e73dba0a96257b+'<\/a>';

The McKay matrix $M_V$ records the result of tensoring the simple modules with a fixed finite-dimensional module $V$. It is the adjacency matrix of the quiver determined by tensoring with $V$. For a finite group, the eigenvectors for $M_V$ are the columns of the character table, and the eigenvalues come from evaluating the character of $V$ on conjugacy class representatives.

Tensoring determines a Markov chain, and the McKay matrix $M_V$ is closely related to the transition matrix, which tells us the probability of going from one site to another on the chain. We describe two different Markov chains obtained from taking tensor products of the Weyl modules with the two-dimensional Weyl module $V$ for the quantum group $U_q(sl_2)$, when $q^2$ is a primitive $\ell$th root of unity for $\ell$ an odd integer $\ge 3$.

Joint work with Samuel A. Lopes (Universidade do Porto, Portugal).