The structure of the Garsia-Procesi modules $R_\mu$

Garsia and Procesi defined the sets of monomials $\mathcal{B}_\mu$ recursively by:

$$\mathcal{B}_{\square}=\{1\},$$ $$\mathcal{B}_{\mu}=\bigcup_d x_n^{d-1} \mathcal{B}_{\mu^{(d)}}$$ where $n=|\mu|$ and $\mu^{(d)}$ is the smaller partition formed by removing the corner square that is in column $\mu_d$. They also showed that $\mathcal{B}_\mu$ forms a basis of $R_\mu$ for each $\mu$.


This set can also be described explicitly (see my thesis) in terms of Yamanouchi words. A word $w$ with $\mu_i$ copies of $i-1$ for each $i$ is said to be Yamanouchi if every suffix of the word also has partition content: that is, every tail $w_k,\ldots,w_n$ has at least as many $i-1$’s as $i$’s for each $i$. For instance, $1020010$ is Yamanouchi, but $0100120$ is not because the suffix $20$ has more $2$’s than $1$’s. The partition $\mu$ is called the content of the word; as an example, the word $1020010$ has content $(4,2,1)$ since there are four $0$’s, two $1$’s, and one $2$ in the word. Let $\mathrm{Yam}(\mu)$ be the set of all Yamanouchi words of content $\mu$.

For a word $w$, define $x^w=x_n^{w_1}\cdots x_1^{w_n}$. Then $$\mathcal{B}_\mu=\{x^d:\exists w\in \mathrm{Yam}(\mu), x^d|x^w\}$$ is the set of monomials that divide some monomial of a Yamanouchi word of content $\mu$.

The fascinating thing is that this lower order ideal of monomials has size $\binom{n}{\mu}$. This is not immediately obvious, especially since some of the monomials divide more than one Yamanouchi monomial. But one nice way of counting them is by interpreting the degrees of these monomials as generalized Carlitz codes, and tying them to the Macdonald $\mathrm{inv}$ statistic.

First, note that in the case $\mu=(1^n)$, the only Yamanouchi word $w$ of content $\mu$ is $n-1,n-2,\ldots,3,2,1,0$, and so the monomials that divide $x^w$ are precisely those whose degrees form a Carlitz code, as described in this post.

Indeed, consider the Macdonald fillings with distinct entries $1,\ldots, n$ of the transpose $\mu’$ with $\mathrm{maj}(\sigma)=0$, as described in this post (we will be using all the notation from that post from here on). This condition means that all the columns are weakly increasing when read from top to bottom, and there are no other restrictions on the filling. The $\mathrm{inv}$ statistic on such tableaux can be computed by counting the number of attacking pairs, since there are no descents. If we assign to an entry $i$ the number $a(i)$ of attacking pairs that involve $i$ as the smaller entry, then it turns out that the resulting monomials $\prod x_{n-i}^{a(i)}$ are exactly the monomials of $\mathcal{B}_\mu$.

Since there are exactly $\binom{n}{\mu}$ ways to choose which entries go in each column of our filling, and only one way to make the columns increasing once we made that choice, it’s an immediate consequence that $|\mathcal{B}_\mu|=\binom{n}{\mu}$.

2 thoughts on “The structure of the Garsia-Procesi modules $R_\mu$

  1. I like your post! One thing that could be confusing to readers that you may want to point out is that the notation O(\overline{C_{\mu’}}\cap T) means we are taking the scheme-theoretic intersection of the two varieties, rather than the set-theoretic intersection. In fact, the intersection of the two varieties as sets is just {0}, because the only diagonal matrix which is nilpotent is the zero matrix! What you really want to do is take the sum of the defining ideals of the two varieties and quotient out by that new ideal.

    You might also need to be careful when you say “if X=diag(x_1,\dots,x_n)\in \overline{C_{\mu’}}\cap T…”, for the same reasons.

    • Great point, thanks for your comment! I’ll just leave your comment here rather than changing the post so that the comment still makes sense. 🙂

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