--- title: Exotic Spaltenstein Varieties url: https://www.emergentmind.com/topics/exotic-spaltenstein-varieties type: topic --- # Exotic Spaltenstein Varieties Exotic Spaltenstein varieties are a family of complex algebraic varieties that generalize Spaltenstein varieties—partial flag analogues of Springer fibres—to the context of exotic Springer theory. Defined over the complex numbers and related to the symplectic group $\mathrm{Sp}_{2n}$, these varieties are constructed from data involving self-adjoint nilpotent endomorphisms and the so-called exotic nilpotent cone. Their geometry is governed by intricate combinatorics involving semi-standard Young bitableaux, and their irreducible components and dimensions are controlled by combinatorial formulas. Exotic Spaltenstein varieties have deep connections to the exotic Robinson–Schensted–Knuth (RSK) correspondence and are conjecturally linked to a geometric framework for representation theory and combinatorics [2410.00235]. ## 1. Construction and Definition Let $V$ be a $2n$-dimensional complex vector space equipped with a nondegenerate skew-symmetric bilinear form $(\cdot,\cdot)$. The symplectic group is defined as \[ \mathrm{Sp}_{2n} = \{g \in \mathrm{GL}(V) \mid (g v, g w) = (v, w) \ \forall\, v, w \in V\}. \] Let $N \subset \mathrm{End}(V)$ denote the nilpotent cone, and let $S \subset \mathrm{End}(V)$ be the space of self-adjoint endomorphisms, i.e., those with $(v, x w) = (x v, w)$ for all $v, w \in V$. The exotic nilpotent cone is then $\mathfrak{N} := V \times (S \cap N)$, with $\mathrm{Sp}_{2n}$ acting diagonally. Given a composition $a = (a_1, ..., a_m)$ of $n$, the “palindromic” composition \[ \alpha = (a_1, a_2, ..., a_m, a_m, ..., a_2, a_1) \] of $2n$ is defined, and $d_i = \sum_{j=1}^i \alpha_j$ ($d_0=0$, $d_{2m}=2n$). The isotropic partial flag variety of type $\alpha$ is \[ \mathcal{F}^\alpha(V) = \{0 = F_0 \subset F_1 \subset \dots \subset F_{2m} = V \mid \dim F_i - \dim F_{i-1} = \alpha_i,\, F_i \subset F_i^\perp\}. \] Given $(v, x) \in \mathfrak{N}$, the exotic Spaltenstein variety of type $\alpha$ over $(v, x)$ is \[ C^\alpha(v, x) = \{F_\bullet \in \mathcal{F}^\alpha(V) \mid v \in F_n,\ x(F_i) \subset F_{i-1}\ \forall\, i\}. \] This construction generalizes the exotic Springer fibre, with $\alpha = (1, 2, \ldots, n, n, n-1, ..., 1)$ corresponding exactly to Kato’s exotic Springer fibre [2410.00235]. ## 2. Self-adjoint Nilpotent Endomorphisms of Order Two Suppose $x \in S \cap N$ satisfies $x^2 = 0$ but $x \neq 0$. The Jordan normal form of $x$ on $V$ is $(2^{n_2}, 1^{n_1})$ with $n_1 + n_2 = n$. In the symplectic context, the exotic Jordan type is a bipartition $(\mu, \nu) \in \mathcal{Q}_n$ with $\mu + \nu = (n_2, n_2, n_1)$. There exists a normal basis with explicit pairing and $x$-action structure: \[ (v_{i,j}, w_{i',j'}) = \delta_{i,i'} \delta_{j + j', \mu_i + \nu_i + 1}, \quad x v_{i,j} = v_{i,j-1}, \quad x w_{i,j} = w_{i,j-1}. \] The vector $v$, associated to the "wall" separating the $\mu$ from the $\nu$ part, determines further structure in this setup [2410.00235]. ## 3. Top-dimensional Irreducible Components and Semi-standard Young Bitableaux For $eJ(v,x)=(\mu,\nu)$, consider a flag $F_\bullet \in C^{\alpha}(v,x)$. At each step, examine the symplectic quotient $(F_i/F_{i-1}, x)$, whose exotic Jordan type is $(\mu^{(i)}, \nu^{(i)})$. The structure induces a combinatorial object: a semi-standard Young bitableau of shape $(\mu, \nu)$ and content $\alpha$: \[ (\mu^{(0)}, \nu^{(0)}) = (\varnothing, \varnothing) \longrightarrow \cdots \longrightarrow (\mu^{(2m)}, \nu^{(2m)}) = (\mu, \nu) \] where, for each $i$, $\alpha_i$ boxes are added in a vertical strip (no two in the same row of either diagram). Entries strictly increase in rows and weakly increase in columns, with filling rules determined by the geometry. Let $\mathrm{SSYT}^{(\mu, \nu)}_\alpha$ denote the set of such bitableaux. There is a map \[ \Phi: \{\text{irreducible components of } C^{\alpha}(v, x)\} \to \bigsqcup_{(\mu, \nu) = eJ(v, x)} \mathrm{SSYT}^{(\mu, \nu)}_\alpha \] which, for $x^2 = 0$, induces a bijection on the set of top-dimensional irreducible components. Specifically, all top-dimensional irreducible components of $C^{\alpha}(v, x)$ are in bijection with $\mathrm{SSYT}^{(\mu, \nu)}_\alpha$, and every other component corresponds to a non-semistandard sequence and hence is of strictly lower dimension [2410.00235]. ## 4. Combinatorial Formula for the Top Dimension Given a bipartition $(\mu, \nu)$ and composition $\alpha$, define \[ N(\lambda) := \sum_{i} (i-1) \lambda_i \quad \text{for any partition } \lambda, \] \[ d_{\mu, \nu}^\alpha := 2 N(\mu + \nu) + |\nu| - \sum_{i=1}^m (\alpha_i - 1), \] where $\mu + \nu$ is concatenation and $|\nu| = \sum \nu_i$. Then \[ \dim_\mathrm{top}\, C^{\alpha}(v, x) = d_{\mu, \nu}^\alpha. \] In the special case $\alpha = (1, ..., 1)$, this reduces to the formula for Kato’s exotic Springer fibre dimension, $2N(\mu+\nu)+|\nu|$ [2410.00235]. ## 5. Conjectural Extension to Arbitrary Nilpotent Order Rosso–Saunders conjecture that for any nilpotent $x \in S$ with exotic Jordan type $(\mu, \nu)$, all top-dimensional irreducible components of $C^\alpha(v, x)$ are in bijection with $\mathrm{SSYT}^{(\mu, \nu)}_{\alpha}$ via $\Phi$, each with dimension $d_{\mu, \nu}^\alpha$, and every component of lower dimension corresponds to a non-semistandard sequence. The $x^2 = 0$ case provides the first nontrivial instance of this conjecture; higher nilpotency order remains open [2410.00235]. ## 6. Exotic RSK–Knuth Correspondence and Steinberg Varieties In analogy with the geometric RSK correspondence for pairs of partial flags, the exotic setting introduces the exotic Steinberg variety \[ Z^{\alpha, \beta} = \{(F_\bullet, G_\bullet, (v, x)) \mid F_\bullet \in C^{\alpha}(v, x),\, G_\bullet \in C^{\beta}(v, x)\}, \] with stratification by the $\mathrm{Sp}_{2n}$-orbit of $(F_\bullet, G_\bullet)$. The relative position is encoded by a $2m \times 2\ell$ palindromic matrix $A$ with nonnegative entries, row sums $\alpha$, column sums $\beta$. Each stratum is conjectured to be pure of dimension \[ 2 n^2 - \sum_{i