Q-data and Representation theory of untwisted quantum affine algebras

  • Ryo Fujita
  • ,
  • Se-jin Oh

For a finite-dimensional complex simple Lie algebra $\mathfrak{g}$, we
introduce the notion of Q-datum, which generalizes the notion of Dynkin quiver
with height function from the viewpoint of Weyl group combinatorics. Using this
notion, we develop a unified theory for Auslander-Reiten quivers, twisted
Auslander-Reiten quivers and their repetition quivers with generalized Coxeter
elements. From the development, we obtain the $\mathfrak{g}$-additive property
of the (twisted) Auslander-Reiten quivers, which yields a combinatorial formula
expressing the inverse of quantum Cartan matrix of $\mathfrak{g}$. We also find
a unified expression of the denominators of R-matrices between all the
Kirillov-Reshetikhin modules (except few cases) from these results. In short,
the quivers, the inverse of quantum Cartan matrix and the denominators can be
obtained from any Q-datum for $\mathfrak{g}$. Since the quivers, the inverse of
quantum Cartan matrices and the denominators of R-matrices play important roles
in the finite-dimensional representation theory of untwisted quantum affine
algebra of $\mathfrak{g}$, we have interesting applications. For instances, (i)
we establish complete unified formulae for the universal coefficients between
all the fundamental modules, (ii) we recover the main results of
[Kashiwara-Kim-O-Park, arXiv:2003.03265] in a unified way, (iii) we compute the
invariant $\Lambda(M,N)$ for a commuting pair of simple modules $M$ and $N$ in
terms of the anti-symmetric pairing arising from the quantum tori introduced by
Nakajima, Hernandez and Varagnolo-Vasserot.

Arxiv Url
Arxiv Url
http://arxiv.org/pdf/2007.03159v1 本文へのリンクあり