CharacterVarieties.jl

This package computes $E$-polynomials of character varieties associated to general reductive groups in Julia. This was written for the paper arXiv:2409.04735. We heavily rely on Jean Michel's port of the computer algebra system Chevie.

Getting started

Download and install Julia. In the REPL (Julia's interactive command-line), copy-paste and run the below:

using Pkg; Pkg.add(url="https://github.com/baileywhitbread/CharacterVarieties.jl")

This will install the CharacterVarieties.jl package and its dependencies. To load the package, copy-paste and run the below:

using CharacterVarieties

Multiplicative and additive character varieties

Fix integers $g\geq 0$ and $n\geq 1$ and let $G$ be a connected split reductive group over $\mathbb{F}_q$ with connected centre $Z$ and split maximal torus $T$. Select strongly regular elements $S_1,\ldots,S_n$ in $T$ that are 'generic' and let $C_1,\ldots,C_n$ be their conjugacy classes. The multiplicative character variety is the GIT quotient

$$\mathbf{X} = \bigg\{(A_1,B_1,\ldots,A_g,B_g,Y_1,\ldots,Y_n)\in G^{2g}\times C_1\times \cdots\times C_n\ \bigg|\ [A_1,B_1]\cdots[A_g,B_g]Y_1\cdots Y_n = 1\bigg\}\bigg/\!\!\!\!\!\bigg/G$$

where the action is simultaneous conjugation. This is an affine scheme of finite type over the finite field of size $q$.

Let $\mathfrak{g}$ be the Lie algebra of $G$ and let $\mathfrak{t}$ be the Lie algebra of $T$. Select regular semisimple elements $s_1,\ldots,s_n$ in $\mathfrak{t}$ that are 'generic' and let $O_1,\ldots,O_n$ be their adjoint orbits. The additive character variety is the GIT quotient

$$\mathbf{Y} = \bigg\{(A_1,B_1,\ldots,A_g,B_g,Y_1,\ldots,Y_n)\in \mathfrak{g}^{2g}\times O_1\times \cdots\times O_n\ \bigg|\ [A_1,B_1]+\cdots+[A_g,B_g] + Y_1+ \cdots + Y_n = 0\bigg\}\bigg/\!\!\!\!\!\bigg/G$$

where the action is simultaneous conjugation (i.e., the adjoint action). This is an affine scheme of finite type over the finite field of size $q$.

Calculating E-polynomials

This package computes the $E$-polynomials $E(\mathbf{X};q)$ and $E(\mathbf{Y};q)$ (see LRV for the definition of $E$-polynomials of varieties over finite fields or HRV for the complex analogue). This is done using our formulas for $E(\mathbf{X};q)$ and $E(\mathbf{Y};q)$.

We will use the semisimple group of adjoint type $G_2$ as an example. The command G=coxgroup(:G,2) selects this group.

One can instead choose coxgroup(:A,2), coxgroup(:B,2), and so on. Alternatively, one can select these groups using rootdatum(:pgl,3) or rootdatum(:so,5), or non-semisimple groups such as rootdatum(:gl,2).

Then EX(G,g,n) returns $E(\mathbf{X};q)$, and EY(G,g,n) returns $E(\mathbf{Y};q)$.

For example:

julia> EX(G,0,3)
Pol{BigInt}: q⁸+6q⁷+20q⁶+58q⁵+180q⁴+58q³+20q²+6q+1

julia> EY(G,0,3)
Pol{BigInt}: q⁸+6q⁷+19q⁶+45q⁵+99q⁴

Further directions

Associated to $\mathbf{X}$ is the (compactly supported) mixed Hodge polynomial

$$H(\mathbf{X};x,y,t) = \sum_{i,j,k} h^{i,j,k} x^i y^j t^k.$$

Proving $\mathbf{X}$ is polynomial count implies $H(\mathbf{X};x,y,t)$ depends only on the product $xy=:q$ and $t$, and that

$$H(\mathbf{X};q,-1) = E(\mathbf{X};q).$$

There's another specialisation of $H(\mathbf{X};q,t)$ given by setting all terms to zero except monomials in $u:=qt^2$:

$$H(\mathbf{X};q,t)=qt^2 + q^2t^4 + q + qt + 1 \rightsquigarrow PH(\mathbf{X};u):=u+u^2+1.$$

It is conjectured the polynomials $H(\mathbf{X};q,t)$ and $E(\mathbf{Y};u)$ are closely related, in the sense that

$$PH(\mathbf{X};q) = E(\mathbf{Y};q).$$

This is known in one case (because $H(\mathbf{X};q,t)$ is explicitly known): $G=\mathrm{GL}_2$, $n=1$ and $C_1$ is the conjugacy class of

$$\left(\begin{smallmatrix}-1 & \\ & -1 \end{smallmatrix}\right).$$

When $G=\mathrm{GL}_d$ and the $C_i$ are semisimple, an unproven formula for $H(\mathbf{X};q,t)$ was given in HLRV. One could use the specialisations $E(\mathbf{X};q)$ and $E(\mathbf{Y};q)$ to try and understand the mixed Hodge polynomial $H(\mathbf{X};q,t)$.

The formulas for $E(\mathbf{X};q)$ and $E(\mathbf{Y};q)$ in HLRV are very different to ours. When $G=\mathrm{GL}_d$, the coefficients of $\#\mathbf{Y}(\mathbb{F}_q)$ are non-negative by the work of HLRV. One could use CharacterVarieties.jl to search for $E(\mathbf{Y};q)$ with negative coefficients, or other interesting behavior.