Parametrization of Schubert varieties in isotropic Grassmannians by partitions

Question

Let $X=\mathbb{G}_Q(l,p)$ be the isotropic Grassmannian, where $l\leq p-2$. Let $q=p-l$. Let $W^P$ be the set of minimal length representatives. Let $\tilde{\mathcal{Q}}(l,p)$ be the set of partition pairs $(\alpha,\lambda)$ with type attached to them where $\alpha$ is a partition of shape $q\times l$ and $\lambda$ is a strict partition of shape $l\times(p-1)$, where $\alpha_q\geq\ell(\lambda)$ and where $\mathrm{type}(\alpha,\lambda)=0$ if $\alpha_q=\ell(\lambda)$ and $=1$ or $2$ otherwise.

I want to define a bijection $\varphi$ between $W^P$ and $\tilde{\mathcal{Q}}(l,p)$ in a sensible (this means it should match with the literature cited bellow) way such that the codimension formula or the dimension formula hold, i.e. such that $\ell(w)+|\varphi(w)|=\dim(X)$ or such that $\ell(w)=|\varphi(w)|$.

The question is basically about how to adapt the known construction from [Quantum Pieri rules for isotropic Grassmannians by Buch, Kresch, Tamvakis, Section 4.5] and [Quantum cohomology of isotropic Grassmannians by Tamvakis, Section 6.1] to my notation and my needs.

I exlpain now what I tried so far. The elements of $W^P$ can be described as barred permutations as follows: $$ w=(u_1,\ldots,u_t,\overline{u_{t+1}},\ldots,\overline{u_l},u_{l+1},\ldots,\hat{u}_p)$$ where $0\leq t\leq l$, $u_1<\cdots<u_t$, $u_{t+1}>\cdots>u_{l}$, $u_{l+1}<\cdots<u_p$ and $\hat{u}_p=u_p$ if $l-t$ is even and $=\overline{u_p}$ if $l-t$ is odd. Given such a barred permutation we can define $\lambda_i=u_{t+i}-1$ for all $1\leq i\leq l-t$ and $\alpha_i=u_{p-i+1}+i-q-1+\mathrm{card}\{t<j\leq l\mid u_j>u_{p-i+1}\}$ for all $1\leq i\leq q$. We set $\mathrm{type}(\alpha,\lambda)=0$ if $\alpha_q=\ell(\lambda)$ and otherwise $=1$ if $\hat{u}_p=u_p$ and $=2$ if $\hat{u}_p=\overline{u_p}$. This defines a welldefined bijection but neither the codimension nor the dimension formula hold, for example the identity corresponds to $((l^q),\emptyset)$.

Can anyone please help me to adapt the definition of $\varphi$. (This question is about my research, so there is no reason to remove it from the site.) Thanks a lot!

Answer 1

Let $w$ be a minimal length representative with representation as in the question. Then we can define a map $\psi\colon W^P\to W^P$ by setting $\psi(w)=w'$ where $$w'=(p+1-u_t,\ldots,p+1-u_1,\overline{p+1-u_l},\ldots,\overline{p+1-u_{t+1}}.p+1-u_p,\ldots,\widehat{p+1-u_{l+1}})\,.$$ This is in an involution, in particular bijective. The composite $f=\varphi\circ\psi$ satisfies $\ell(w)=|f(w)|$ for all $w\in W^P$ and is a bijection between $W^P$ and $\tilde{\mathcal{Q}}(l,p)$.

It is still unclear to me (but not subject to the question) which linear spaces are parametrized by such a partition, if they match with the ones given the references cited above.