#
t--analog of q--characters

## Introduction

`Qchar' is the C program for the computation of the -analog of -characters of level fundamental representations of the quantum loop algebra
associated with a
a simple Lie algebra of type .
-analog of -characters were introduced by myself in
- -analogue of the -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 --analogs of --characters of quantum affine algebras, Ann. of Math. 160 (2004), 1057--1097.

There is a combinatorial algorithm to compute for any simple
-module. It is based on the study
of perverse sheaves on graded/cyclic quiver varieties.
The algorithm is roughly separted into three steps:
- Compute for level fundamental modules.
- Compute for standard modules .
- Compute for simple modules .

The program `Qchar' is for the step 1. The step 2 is a twisted
multiplication of of level 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 fundamental module to the
-module.

## Installation

- Download Qchar.tgz
- % tar xzvf Qchar.tgz
- Edit Makefile according to your system.
- % make all

(I have compiled with `gcc'. I do not know about other C's.)
For :
Required Memory

level : 120GByte

level : 2.6GByte

level : 120MByte

The most important parameters are LENMAX and MAX_NUM_TERMS.

## Usage

For example, for a computation of the first level fundamental
module of , we type
% ./E6 1,0,1

In general,

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

computes the starting

But when contains an -dominant monomial other than -highest ones, the answer is not guaranteed. Level fundamental modules are known to have no -dominant monomials other than -highest.
There are various options. Just type

% ./E6

to see the usage.

To get the ordinary character from the q-character,

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

nakajima@math.kyoto-u.ac.jp