The Garsia-Procesi Modules: Part 2

This is a long overdue followup post to Garsia-Procesi Modules: Part 1 that I finally got around to editing and posting. Enjoy!

In this post, I talked about the combinatorial structure of the Garsia-Procesi modules $R_\mu$, the cohomology rings of the type A Springer fibers. Time to dive even further into the combinatorics!

Tanisaki generators, visually

Recall that, for a partition $\mu$ of $n$, the graded $S_n$-module $R_\mu$ can be constructed (due to Tanisaki) as the quotient ring \[\mathbb{C}[x_1,\ldots,x_n]/I_\mu\] where $I_\mu$ is generated by certain partial elementary symmetric functions called Tanisaki generators.

Define $d_k(\mu)=\mu’_{n-k+1}+\mu’_{n-k+2}+\cdots$ to be the sum of the last $k$ columns of $\mu$, where we pad the conjugate partition $\mu’$ with $0$’s in order to think of it as as a partition of $n$ having $n$ parts. Then the partial elementary symmetric function $e_r(x_{i_1},\ldots,x_{i_k})$ is a Tanisaki generator if and only if $k-d_k(\mu)\lt r\le k$. Tanisaki and Garsia and Procesi both use this notation, but I find $d_k$ hard to remember and compute with, especially since it involves adding zeroes to $\mu$ and adding parts of its transpose in reverse order, and then keeping track of an inequality involving it to compute the generators.

Counting ballots with crystals

In my graduate Advanced Combinatorics class last semester, I covered the combinatorics of crystal base theory. One of the concepts that came up in this context was ballot sequences, which are motivated by the following elementary problem about voting:

Suppose two candidates, A and B, are running for local office. There are 100 voters in the town, 50 of whom plan to vote for candidate A and 50 of whom plan to vote for candidate B. The 100 voters line up in a random order at the voting booth and cast their ballots one at a time, and the votes are counted real-time as they come in with the tally displayed for all to see. What is the probability that B is never ahead of A in the tally?

Doing mathematics in a pandemic - Part IV: Talks with OBS

This is the final post in a four-part series on adapting to the pandemic as a mathematician. See Part I – AlCoVE, Part II – Collaboration, and Part III - Teaching.

As conferences moved online, a number of different methods of giving a remote talk became commonplace. One was to simply point a webcam at a chalkboard and lecture as usual. Another is to make slides and use the “Share Screen” option on Zoom to show the slides to the audience. Another popular method, which I have used a number of times, is to make partial handwritten “slides” in Notability or GoodNotes on an iPad, with space left for doing examples and computations, and then share the iPad screen over Zoom and walk the audience through.

Doing mathematics in a pandemic - Part III: Teaching

This is the third post in a four-part series on adapting to the pandemic as a mathematician. See Part I - AlCoVE, Part II - Collaboration, and Part IV - Talks with OBS.

Of all the things I had to figure out how to adapt to the pandemic reality, I found teaching to be the most challenging by far. So much of the value of teaching comes from the in-person connection between students and teachers, and between peers in the classroom. How can you replicate an entire classroom experience on a 14 inch computer screen? How can you pull off hybrid teaching without diminishing the experience for those students who take the course remotely?

Doing mathematics in a pandemic - Part II: Collaboration

This is the second post in a four-part series on adapting to the pandemic as a mathematician. See _Part I – AlCoVE, Part III – Teaching__, and _Part IV – Talks with OBS

The first aspect of academia to be affected by the pandemic was conferences; the second was in-person collaborative projects. That research collaborator you invited to speak in your seminar can no longer visit, and the potential for a two-day intense collaboration to kick off a new project diminishes drastically. You can no longer meet in person with your graduate students, at least not as easily. Little things like deciding when you’re going to hold the fall Putnam club meetings suddenly turn from a quick conversation in the math department hallway into a five-email exchange.

So I, like all other mathematicians, found ways to adapt. I’ll share a few things that really worked, a few things that really didn’t, and a few extra tools that made things nicer.