# t--analog of q--characters

## Introduction

Qchar' is the C program for the computation of the $t$-analog of $q$-characters of level $0$ fundamental representations of the quantum loop algebra ${\mathbf U}_q({\mathbf L}{\mathfrak g})$ associated with a a simple Lie algebra $\mathfrak g$ of type $ADE$. $t$-analog of $q$-characters $\chi_{q,t}$ were introduced by myself in
• $t$-analogue of the $q$-characters of finite dimensional representations of quantum affine algebras, in Physics and Combinatorics'', Proceedings of the Nagoya 2000 International Workshop, World Scientific, 2001, 195--218.
• Quiver varieties and $t$--analogs of $q$--characters of quantum affine algebras, Ann. of Math. 160 (2004), 1057--1097.
There is a combinatorial algorithm to compute $\chi_{q,t}$ for any simple ${\mathbf U}_q({\mathbf L}{\mathfrak g})$-module. It is based on the study of perverse sheaves on graded/cyclic quiver varieties. The algorithm is roughly separted into three steps:
1. Compute $\chi_{q,t}$ for level $0$ fundamental modules.
2. Compute $\chi_{q,t}$ for standard modules $M(P)$.
3. Compute $\chi_{q,t}$ for simple modules $L(P)$.
The program Qchar' is for the step 1. The step 2 is a twisted multiplication of $\chi_{q,t}$ of level $0$ fundamental modules. The step 3 is analog of the definition of Kazhdan-Lusztig polynomials.

The accompanied program q2c' computes the ordinary character of the restriction of the level $0$ fundamental module to the ${\mathbf U}_q(\mathfrak g)$-module.

## Installation

2. % tar xzvf Qchar.tgz
3. Edit Makefile according to your system.
4. % make all
(I have compiled with gcc'. I do not know about other C's.) For $E_8$:

$\begin{matrix}7 & -- & 6 & -- & 5 & -- & 4 & -- & 3 & -- & 2 & -- & 1 \\ &&& & | &&&&&&&&&& \\ &&& & 8 &&&&&&&&&& \end{matrix}$

Required Memory

$5^\mathrm{th}$ level $0$ : 120GByte

$4^\mathrm{th}$ level $0$ : 2.6GByte

$6^\mathrm{th}$ level $0$ : 120MByte

The most important parameters are LENMAX and MAX_NUM_TERMS.

## Usage

For example, for a computation of the first level $0$ fundamental module of $E_6$, we type

% ./E6 1,0,1

In general,

% ./E6 a,b,c,d,e,f,...

computes the $\chi_{q,t}$ starting

$Y_{a,q^b}^c Y_{d,q^e}^f ...$

But when $\chi_{q,t}$ contains an $\ell$-dominant monomial other than $\ell$-highest ones, the answer is not guaranteed. Level $0$ fundamental modules are known to have no $\ell$-dominant monomials other than $\ell$-highest. There are various options. Just type

% ./E6

to see the usage.

To get the ordinary character from the q-character,

1. % ./E6 -f 1,0,1 (This store the date to files.)
2. % ./E6_q2c 1,0,1

nakajima@math.kyoto-u.ac.jp