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Wednesday, September 16, 2015

**Abstract:** T-systems and their combinatorial solutions were introduced in the previous talks. We know T-systems show up in relation to commuting transfer matrices in solvable lattice models, cluster algebras, difference analog of $L$-operators in KP hierarchy, and many more. In this talk, we will discuss T-systems appearing in Representation Theory of quantum affine algebras. In particular, $q$-characters of Kirillov-Reshetikhin modules over quantum affine algebras satisfy the T-system. This result is highly non-trivial and was proved by Nakajima (2003) for types ADE via geometrical construction of quiver varieties and later generalized to all cases by Hernandez. We will introduce the quantum affine algebra $U_q( \widehat{\mathfrak{sl}}_2)$, define $q$-characters, and give some examples. We will then discuss what is known and, more importantly, what is not known about $q$-characters. If time permits, we will broadly introduce $q,t$-characters and the deformed $T$-system they define.