The n-Category Café

Should that last link be this? The one you gave is to the ACT@UCR Zulip chat, not the Category Theory one.

I’m excited that these are getting up and running! I’ve been anticipating these seminars for so long, and it’s genuinely exciting that they’re actually about to start.

Something tells me that the audience in the room will be far outnumbered by the hordes encapsulated in the little Zoom participants window in front of me.

Still no nilpotent orbits, huh? Let’s see if I can get you to take an interest.

Story #1. Fix a partition $\lambda$ of $n$ and let $O$ be the closure of the space of nilpotent matrices whose Jordan blocks are the sizes of the rows of $\lambda$. Define the orbital scheme $O \cap b$ to be its intersection with the upper triangular matrices. Theorem: the smooth part of $latex O$ is (holomorphic) symplectic, and the orbital scheme is Lagrangian therein.

If we consider the NW $k\times k$ submatrix of $M \in O \cap b$, for each $k$, we get another strictly upper triangular matrix… hence nilpotent. This chain of nilpotents gives a chain of partitions, hence, a standard Young tableau of shape $\lambda$.

This function $O \cap b \to SYT_\lambda$ is “constructible”, and consequently, is constant on a dense open set of each irreducible component of $O\cap b$. So we get a function from the set of components to $latex SYT_\lambda$. Theorem (Spaltenstein): that map is bijective.

Okay, how do we define an action of $S_n$ on the top Borel-Moore homology of this space, now that we know it has the right dimension to be an $S_n$ irrep? Of course it’s enough to determine the action of the simple reflections.

Given a component $C$, sweep it out using the $i$th $SL_2$ along the diagonal. The sweep will be one dimension larger, or stay $C$. In the first case, intersect the sweep with $b$ again, at which point, it will be a schemy union of other components $C'$. Put the multiplicities of those components into the $C$ column of $r_i$. There’s a little more fiddling (like subtracting twice the identity matrix) but this is the geometric origin of the representation.

In particular this produces each irrep of $S_n$ exactly once, each with a canonical basis, in which $r_i$ doesn’t act positively but with predictable signs.

Story #2. A complex vector space with a nilpotent operator is almost the same as a $C[[z]]$-module. Fix three such modules $M_\lambda,M_\nu,M_\mu$, corresponding to the partitions $\lambda,\nu,\mu$. Consider the space of short exact sequences $0 \to M_\lambda \to M_\nu \to M_\mu \to 0$. This time, the components of the space don’t count SYT but instead count Littlewood-Richardson tableaux, i.e. their number is $c_{\lambda\mu}^\nu$.

(Of course it’s interesting to replace $C[[z]]$ with the $p$-adics and to count SES instead of take their top homology.)

Hooray! The first This Week’s Finds Seminar is now on YouTube.

In the spirit of sharing, and in case anyone cares, here are the slides for my little introduction.