Post 2: Current Research

23 Dec 2023

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In this post I’ll talk about my current research project.

I work with Pramod Achar (opens in new tab). Our goal is to give explicit models for so-called “global Schubert varieties” for $GL_n$. Let me first give intuition¹ for these words, and then I’ll talk specifics.

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Intuition

Algebraic groups²

Underlying almost all of geometric representation theory are objects called “algebraic groups”. One should note that these are not just the same as groups as you may have seen in an abstract algebra course. Rather, depending on your perspective, they are either a family of groups, or a certain kind of geometric object with a group structure. If you have seen Lie groups before, they are closely related.

The most familiar and motivating example of an algebraic group is \(GL_n\). You may have seen the group $GL_n(\R)$ before; it is the group of $n$ by $n$ invertible matrices with entries in $\R$. You may have also seen that you can consider $GL_n(\C)$ as well. In fact, for any (commutative and unital) ring $R$, there is a corresponding group $GL_n(R)$. We can then consider $GL_n$ to be a single object that takes rings $R$ and gives us groups $GL_n(R)$.

We have arrived at the core idea of algebraic groups: for any ring \(R\), an algebraic group \(G\) has a corresponding group \(G(R)\). There is extra compatibility³ and niceness⁴ that one often imposes, but \(GL_n\) has these properties, and I won’t be mentioning them explicitly. In fact, I’ve essentially described a more general notion, known as a group scheme. In any case, the details aren’t important for what’s to come.

Explicit models

What is an “explicit model”? To put it simply, the objects we are studying are defined in a very arcane way, and we want to find a simpler way to describe it. I don’t really have any good examples of what this means, so hopefully you can piece together what I mean from my own work.

Global Schubert varieties

What is a “global Schubert variety”? Well, there is a lot of background required to even answer that question, even vaguely. So, I will throw out some words and how they relate to each other, so that you have an idea of what I need to define. All of these words can be defined for a sufficiently nice algebraic group, but since I will be fixing $G=GL_n$, I will use articles like “the” instead of “a”.

1. Global Schubert varieties are subspaces of the global affine Grassmannian.

2. The global affine Grassmannian is made out of copies of the affine Grassmannian and the affine flag variety.

3. Global Schubert varieties are defined using Schubert cells and varieties⁵, which are subspaces of the affine Grassmannian.

The affine Grassmannian, denoted \( \mathcal Gr \), is a certain topological space on which the group $GL_n(\C[[t]])$ acts. Here, $\C[[t]]$ is the ring of power series $\sum\limits_{n=0}^\infty c_nt^n$ with coefficients in $\C$. The orbits of this group action are called Schubert cells. Each Schubert cell corresponds to some combinatorial object $\lambda$. For $GL_n$, the possible $\lambda$ are non-increasing sequences of $n$ integers. We denote the cell corresponding to $\lambda$ by $\Gr_\lambda$. A Schubert variety is the topological closure of a Schubert cell in $\Gr$.

In the case \( GL_n \), the affine Grassmannian can be identified with the set of objects called lattices. I won’t describe what lattices are here, but the important thing is that they are hands-on objects that we can use to give meaning to a point in $\Gr$. In other words, lattices allow us to give an explicit model for the affine Grassmannian of \( GL_n \). It is well-known (to geometric representation theorists) what the conditions on a lattice are for it to be in the Schubert cell or variety corresponding to $\lambda$.

The affine flag variety, \( \mathcal Fl \), is similar; it is a certain space associated to our algebraic group that also has a natural group action on it. For \( GL_n \), we can identify \( \mathcal Fl \) with the set of lattice chains. These are lists of \( n \) lattices with some extra conditions.

Finally, the global affine Grassmannian \( \mathbf{Gr} \) is built out of \( \mathcal Gr \) and \( \mathcal Fl \) in the following way: There is a continuous map \(p:\mathbf{Gr}\to \mathbb C\) such that

\[p^{-1}(0)\cong \mathcal Fl,\] \[p^{-1}(\mathbb C^\times) \cong \mathbb C^\times \times \mathcal Gr.\]

Here, \( \mathbb C^\times \) is the set of non-zero complex numbers.

Now is the hilarious part: even though I’ve been working on this project for almost two years, we only realized this month that our explicit model for \(\mathbf{Gr}\) was incorrect. Let me elaborate. Because of the definition of \(\mathbf{Gr}\), we want some kind of object that depends on a complex number \(y\) and interpolates between lattices, when \(y\neq0\), and lattice chains, when \(y=0\). We originally used something called “\(y\)-lattice chains”, but it turns out this gave us results that did not match up with pre-existing (and correct) results.

At the moment, my advisor claims to have found the correct explicit model. Instead of $y$-lattice chains, we use something that we call a “lattice family”, which consists of a complex number and \(n\) lattices satisfying some extra conditions. However, I won’t elaborate on these until we have the details set in stone.

Anyway, now I can finally (loosely) define the global Schubert varieties. Under the isomorphism \(p^{-1}(\mathbb C^\times)\cong \mathbb C^\times\times \mathcal Gr \), we can consider \(\mathbb C^\times\times\mathcal Gr_\lambda\) as a subset of \(\mathbf{Gr}\). Its topological closure is what we call a global Schubert variety.

To reiterate, the goal of our project is to determine the conditions on a lattice family⁶ that tell you whether or not it is in a given global Schubert variety. To that end, we are almost done.

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Formal stuff

Now we go back through all of the previous talk with some actual math. While I tried to make the previous section undergrad-friendly, I make no attempt to do so in this section. In particular, I will use some buzz words that may mean nothing to an undergrad student, especially one that hasn’t done much representation theory. However, if one ignores the fancy lingo and follows the math, it actually isn’t too bad. One still only needs first courses in algebra and topology (I think). So, if you are reading this as an undergrad - try your best!

Lattices and affine Grassmannians

We will use the power series ring \(\mathbb O = \mathbb C[[t]]\) and its fraction field, the Laurent series \(\mathbb K = \mathbb C((t))\). Also, fix positive integer \(n\).

A lattice is a free rank \(n\) \(\mathbb O\)-submodule of \(\mathbb K^n\). In particular, a lattice can be given by a \(\mathbb K\)-basis for \(\mathbb K^n\); it is the \(\mathbb O\)-span of that basis. The standard lattice is \(\mathbb O^n\), which corresponds to the standard basis \(e_1,\dots,e_n\).

The affine Grassmannian of an algebraic group \(G\) is the quotient \(G(\mathbb K)/G(\mathbb O)\). In the case \(G=GL_n\), this can be identified with the set of lattices. The reason for this is the orbit-stabilizer theorem: the group \(GL_n(\mathbb K)\) acts transitively on the set of lattices, and the stabilizer of any particular lattice is isomorphic to \(GL_n(\mathbb O)\). In fact, the stabilizer of \(\mathbb O^n\) is exactly \(GL_n(\mathbb O)\).

By construction, there is a natural action of \(G(\mathbb O)\) on \(\mathcal Gr\). The Schubert cells \(\Gr_\lambda\) are the \(G(\O)\)-orbits, and they are labeled by dominant coweights \(\lambda\) of \(G\). I won’t explain what (dominant) coweights are in general, nor do you need to know what they are. For \(GL_n\), dominant coweight has a simple meaning⁷: it is a non-increasing sequence of \(n\) integers, written as \(\lambda=(\lambda_1\geq\dots\geq\lambda_n)\).

For each dominant coweight $\lam$, there is a convenient representative for $\Gr_\lam$. Let $\L^\lam$ be the lattice corresponding to the basis $t^{\lam_1}e_1,\dots,t^{\lam_n}e_n$. Then $\Gr_\lam$ is the $GL_n(\O)$-orbit of $\L^\lam$.

This definition of $\Gr_\lam$ is not hard to state, but at first glance it doesn’t tell you much. If I give you a lattice $\L$, how can you tell if it is in $\Gr_\lam$ for some dominant coweight $\lam$? Luckily, there is a very nice description:

\[\Gr_\lam=\left\{ \L\in\Gr\mid t^{\lam_1}\O^n\subset \L \subset t^{\lam_n}\O^n \text{ and } \dim\frac{\L\cap t^i\O^n}{t^{\lam_1}\O^n}=\sum_{j=i}^{\lam_1-1}|\{ k\mid \lam_k\leq j \}|\right\}.\]

The second condition must hold for all integers $i$ satisfying $\lam_n\leq i\leq \lam_1$.

We also care about the Schubert variety associated to $\lam$, $\ol{\Gr_\lam}$. Once again, there is a nice description of this space:

\[\ol{\Gr_\lam}=\left\{ \L\in\Gr\mid t^{\lam_1}\O^n\subset \L \subset t^{\lam_n}\O^n \text{ and } \dim\frac{\L\cap t^i\O^n}{t^{\lam_1}\O^n}\ge\sum_{j=i}^{\lam_1-1}|\{ k\mid \lam_k\leq j \}|\right\}.\]

The second condition must hold for all integers $i$ satisfying $\lam_n\leq i\leq \lam_1$.

Lattice chains and affine flag varieties

The assignment $t\mapsto 0$ induces a ring homomorphism $\O\to \C$, and that induces a map $\pi:G(\O)\to G(\C)$. Pick a Borel subgroup $B$ of $G$, and let $I=\pi^{-1}(B(\C))$. $I$ is called an Iwahori subgroup. The affine flag variety $\Fl$ of $G$ is then $G(\K)/I$.

If you don’t know what a Borel subgroup is, that’s fine. For $G=GL_n$, we pick $B$ to be the subgroup consisting of lower triangular matrices. That is, for each ring $R$, $B(R)$ is the group of invertible lower triangular matrices with entries in $R$. Correspondingly, $I$ will consist of elements in $GL_n(\O)$ that are “lower triangular mod $t$”.

Furthermore, we can identify $\Fl$ with the set of lattice chains. A lattice chain $\ul\L$ is a nested sequence of $n$ lattices $L_1\supset L_2\supset\cdots \supset L_n$, with the further conditions that $L_n\supset tL_1$ and $\dim(L_i/L_{i+1})=1$ for $i= 1,\dots,n-1$. It also follows that $\dim(L_n/tL_1)=1$. The reasoning for the identification is the same as that with identifying $\Gr$ with the set of lattices. In particular, $I$ is the stabilizer of the standard lattice chain:

\[\O^n\supset t\O\oplus \O^{n-1}\supset\cdots\supset t\O^{n-1}\oplus\O.\]

The end and global affine Grassmannian

Now, correctly defining the global affine Grassmannian $\bGr$ is beyond my own capability. You can find the original construction in Dennis Gatisgory’s paper “Construction of central elements in the affine Hecke algebra via nearby cycles” (opens in new tab), or you can find a more expository (but still difficult) treatment in the in-progress book of Pramod Achar and Simon Riche, “Central Sheaves on Affine Flag Varieties” (opens in new tab and downloads pdf).

In any case, I will say that the core idea of $\bGr$ is that which I described in the previous section. Namely, it has a map to $\C$, such that over any non-zero complex number it looks like $\Gr$, and over $0$ it looks like $\Fl$.

Since our explicit model of $\bGr$ is in-progress, I am going to end the discussion here. However, I will definitely continue this post when we have checked our work!

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