# Openly available software to work with Demazure modules

Does someone know of any sort of software openly available online which can be used to compute various characteristics of Demazure modules for semisimple Lie algebras? Specifically, I'm interested in dimensions of Demazure modules for type $$A$$. (To clarify: I have my own implementation of the Demazure character formula, I, however, would like something which has been made public in order to refer to it in a paper.)

• Commented Mar 13, 2019 at 7:30
• Ask it at scicomp.stackexchange.com. Commented Mar 13, 2019 at 10:54
• @CarloBeenakker Thank you! Looks promising. Commented Mar 13, 2019 at 11:52
• @user64494 Thank you for the suggestion. I'm pretty sure people are more likely to have an answer for me here though, since this is a community of professional mathematicians. Commented Mar 13, 2019 at 11:56

> bash-3.2\$ LiE
>
> LiE version 2.2.2 created on Oct 22 2018 at 11:36:00 Authors: Arjeh M.
> Cohen, Marc van Leeuwen, Bert Lisser. Purpose: development CWI
>
>
> type '?help' for help information type '?' for a list of help entries.
> > p=Demazure(X[1,1],[1,2],A2)
> > p
1X[-1, 2] +1X[ 0, 0] +1X[ 1,-2] +1X[ 1, 1] +1X[ 2,-1]
> > dim(p,Lie_group(0,2))
>      5
> > quit
> end program

• Thank you! Over the years of using the online demonstration of LiE for dimensions and characters of irreps I totally forgot that there's something much larger behind it! Per's answer is useful in its own right but I'll accept this one since it's closer to the particular thing I was asking for. Commented Mar 13, 2019 at 11:43

Technically, Sage has support to compute type A Demazure characters. The idea is to use the more general function for computing non-symmetric Macdonald polynomials in type A, and then just let $$t=q=0$$. These polynomials are also known as key polynomials.

Also, this page suggests that key polynomials will eventually be implemented directly in Sage as well.

• Thanks! By the way, recently I ran into a bit of a terminology conflict in the literature. Many sources use the term "Demazure polynomial" to denote the same thing as your "key polynomial", i.e. a type A Demazure character. But others (including en.wikipedia.org/wiki/Demazure_module) use it to denote a polynomial in the highest weight expressing the dimension of a Demazure module. What's your take on this, have you ever encountered this second terminology? Commented Mar 13, 2019 at 12:07
• Well, there is the non-symmetric Macdonald polynomials for other types (see e.g., Ram and Yip for a combinatorial formula), and these are expressed using the terminology of roots and highest weight. I am not an expert on the 'root' side of these things, I would love to see a good reference that connected the two. Commented Mar 13, 2019 at 20:23