Exotic Spaltenstein Varieties

• Exotic Spaltenstein varieties are complex algebraic varieties that generalize partial flag analogues of Springer fibres by using self-adjoint nilpotent endomorphisms and the exotic nilpotent cone.
• Their geometry is precisely characterized by semi-standard Young bitableaux and combinatorial formulas that determine the dimensions and irreducible components.
• They play a pivotal role in exotic Springer theory and the exotic RSK–Knuth correspondence, offering new geometric insights into representation theory and combinatorics.

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 (Rosso et al., 2024).

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 $V$0 acting diagonally.

Given a composition $V$1 of $V$2, the “palindromic” composition

$V$3

of $V$4 is defined, and $V$5 ($V$6, $V$7). The isotropic partial flag variety of type $V$8 is

$V$9

Given $2n$0, the exotic Spaltenstein variety of type $2n$1 over $2n$2 is

$2n$3

This construction generalizes the exotic Springer fibre, with $2n$4 corresponding exactly to Kato’s exotic Springer fibre (Rosso et al., 2024).

2. Self-adjoint Nilpotent Endomorphisms of Order Two

Suppose $2n$5 satisfies $2n$6 but $2n$7. The Jordan normal form of $2n$8 on $2n$9 is $(\cdot,\cdot)$0 with $(\cdot,\cdot)$1. In the symplectic context, the exotic Jordan type is a bipartition $(\cdot,\cdot)$2 with $(\cdot,\cdot)$3. There exists a normal basis with explicit pairing and $(\cdot,\cdot)$4-action structure: $(\cdot,\cdot)$5 The vector $(\cdot,\cdot)$6, associated to the "wall" separating the $(\cdot,\cdot)$7 from the $(\cdot,\cdot)$8 part, determines further structure in this setup (Rosso et al., 2024).

3. Top-dimensional Irreducible Components and Semi-standard Young Bitableaux

For $(\cdot,\cdot)$9, consider a flag $$\mathrm{Sp}_{2n} = \{g \in \mathrm{GL}(V) \mid (g v, g w) = (v, w) \ \forall\, v, w \in V\}.$$0. At each step, examine the symplectic quotient $$\mathrm{Sp}_{2n} = \{g \in \mathrm{GL}(V) \mid (g v, g w) = (v, w) \ \forall\, v, w \in V\}.$$1, whose exotic Jordan type is $$\mathrm{Sp}_{2n} = \{g \in \mathrm{GL}(V) \mid (g v, g w) = (v, w) \ \forall\, v, w \in V\}.$$2. The structure induces a combinatorial object: a semi-standard Young bitableau of shape $$\mathrm{Sp}_{2n} = \{g \in \mathrm{GL}(V) \mid (g v, g w) = (v, w) \ \forall\, v, w \in V\}.$$3 and content $$\mathrm{Sp}_{2n} = \{g \in \mathrm{GL}(V) \mid (g v, g w) = (v, w) \ \forall\, v, w \in V\}.$$4: $$\mathrm{Sp}_{2n} = \{g \in \mathrm{GL}(V) \mid (g v, g w) = (v, w) \ \forall\, v, w \in V\}.$$5 where, for each $$\mathrm{Sp}_{2n} = \{g \in \mathrm{GL}(V) \mid (g v, g w) = (v, w) \ \forall\, v, w \in V\}.$$6, $$\mathrm{Sp}_{2n} = \{g \in \mathrm{GL}(V) \mid (g v, g w) = (v, w) \ \forall\, v, w \in V\}.$$7 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 $$\mathrm{Sp}_{2n} = \{g \in \mathrm{GL}(V) \mid (g v, g w) = (v, w) \ \forall\, v, w \in V\}.$$8 denote the set of such bitableaux. There is a map

$$\mathrm{Sp}_{2n} = \{g \in \mathrm{GL}(V) \mid (g v, g w) = (v, w) \ \forall\, v, w \in V\}.$$9

which, for $N \subset \mathrm{End}(V)$0, induces a bijection on the set of top-dimensional irreducible components. Specifically, all top-dimensional irreducible components of $N \subset \mathrm{End}(V)$1 are in bijection with $N \subset \mathrm{End}(V)$2, and every other component corresponds to a non-semistandard sequence and hence is of strictly lower dimension (Rosso et al., 2024).

4. Combinatorial Formula for the Top Dimension

Given a bipartition $N \subset \mathrm{End}(V)$3 and composition $N \subset \mathrm{End}(V)$4, define

$N \subset \mathrm{End}(V)$5

$N \subset \mathrm{End}(V)$6

where $N \subset \mathrm{End}(V)$7 is concatenation and $N \subset \mathrm{End}(V)$8. Then

$N \subset \mathrm{End}(V)$9

In the special case $S \subset \mathrm{End}(V)$0, this reduces to the formula for Kato’s exotic Springer fibre dimension, $S \subset \mathrm{End}(V)$1 (Rosso et al., 2024).

5. Conjectural Extension to Arbitrary Nilpotent Order

Rosso–Saunders conjecture that for any nilpotent $S \subset \mathrm{End}(V)$2 with exotic Jordan type $S \subset \mathrm{End}(V)$3, all top-dimensional irreducible components of $S \subset \mathrm{End}(V)$4 are in bijection with $S \subset \mathrm{End}(V)$5 via $S \subset \mathrm{End}(V)$6, each with dimension $S \subset \mathrm{End}(V)$7, and every component of lower dimension corresponds to a non-semistandard sequence. The $S \subset \mathrm{End}(V)$8 case provides the first nontrivial instance of this conjecture; higher nilpotency order remains open (Rosso et al., 2024).

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

$S \subset \mathrm{End}(V)$9

with stratification by the $(v, x w) = (x v, w)$0-orbit of $(v, x w) = (x v, w)$1. The relative position is encoded by a $(v, x w) = (x v, w)$2 palindromic matrix $(v, x w) = (x v, w)$3 with nonnegative entries, row sums $(v, x w) = (x v, w)$4, column sums $(v, x w) = (x v, w)$5. Each stratum is conjectured to be pure of dimension

$(v, x w) = (x v, w)$6

with irreducible components parameterized by

$(v, x w) = (x v, w)$7

This correspondence generalizes the classical RSK; the matrix/bitableaux bijection is termed the exotic RSK–Knuth correspondence. For $(v, x w) = (x v, w)$8 it recovers the exotic RSK of Nandakumar–Rosso–Saunders (2018) (Rosso et al., 2024).

7. Illustrative Examples

• Exotic Spaltenstein Varieties for $(v, x w) = (x v, w)$9. Here $v, w \in V$0, $v, w \in V$1, and $v, w \in V$2, the standard isotropic partial flag variety. Its dimension matches $v, w \in V$3, and it has a unique irreducible component associated to the unique semi-standard bitableau of shape $v, w \in V$4.
• Case $v, w \in V$5, $v, w \in V$6 of Jordan type $v, w \in V$7. With $v, w \in V$8, $v, w \in V$9, and $\mathfrak{N} := V \times (S \cap N)$0, for $\mathfrak{N} := V \times (S \cap N)$1 ($\mathfrak{N} := V \times (S \cap N)$2 palindromic), $\mathfrak{N} := V \times (S \cap N)$3 is an exotic Springer fibre of full flags, with two irreducible components of dimension $\mathfrak{N} := V \times (S \cap N)$4, in bijection with the two standard Young bitableaux of shape $\mathfrak{N} := V \times (S \cap N)$5:

$\mathfrak{N} := V \times (S \cap N)$6

More generally, exotic Spaltenstein varieties reveal subtle geometry in the partial flag setting, but the structure of their top strata is governed universally by the semi-standard bitableaux associated to their bipartition and composition data (Rosso et al., 2024).