Tymoczko's Dot Action in Equivariant Cohomology

• Tymoczko’s dot action is a Weyl/symmetric-group mechanism that simultaneously reindexes fixed points and permutes Chern-root variables in equivariant cohomology.
• It is naturally formulated in the GKM model and interpreted via monodromy in settings like regular semisimple Hessenberg varieties and quiver Grassmannians.
• Extensions to non-type-A cases reveal rich structures linking Schubert calculus, chromatic quasisymmetric functions, and spline modules in types B, C, and G2.

Searching arXiv for papers on Tymoczko's dot action and related GKM/Hessenberg/quiver Grassmannian work. Tymoczko’s dot action is a Weyl-group, or in type $A$ a symmetric-group, action on equivariant and ordinary cohomology that is most naturally expressed in the GKM model by simultaneously permuting fixed points and polynomial variables. In the type-$A$ flag variety and in regular semisimple Hessenberg varieties, it acts on localized equivariant classes by reindexing vertices of the moment graph and permuting the corresponding Chern-root variables; in ordinary cohomology it descends after setting the equivariant parameters to zero. The same formal pattern extends beyond the classical flag-variety setting to quiver Grassmannians for the equioriented cycle and to spline modules attached to signed-permutation Weyl groups in types $B$ and $C$, while in general reductive type it can also be described as a monodromy action (Lanini et al., 2021, Brosnan et al., 2015, Xue, 2020, Lesnevich, 17 Nov 2025).

1. Equivariant definition in type $A$

For the full flag variety $\mathcal{F}\ell_n = GL_n/B$, let $T \cong (\mathbb{C}^*)^n$ be the usual maximal torus of diagonal matrices. Its equivariant cohomology ring of a point is identified with

$$H_T^*(\mathrm{pt}) = \mathrm{Sym}(X^*(T)\otimes \mathbb{Q}) \cong \mathbb{Q}[x_1,\dots,x_n],$$

where $x_i$ is the Chern root corresponding to the $i$-th coordinate line. Equivariant localization embeds

$A$0

sending a class $A$1 to its collection of fixed-point values $A$2 (Lanini et al., 2021).

On polynomials, the dot action of $A$3 is

$A$4

Equivalently, $A$5 permutes the weights, or Chern roots, on $A$6. If $A$7 is represented by its localizations $A$8, then Tymoczko’s dot action is defined by

$A$9

where $B$0 on the right acts on the polynomial by permuting variables (Lanini et al., 2021).

This definition isolates two simultaneous operations: a permutation of fixed points and a permutation of polynomial coordinates. That pairing is the essential structural feature of the dot action and remains visible in later generalizations.

2. GKM reformulation

The GKM-theoretic formulation expresses the action intrinsically in terms of a moment graph. If $B$1 is a GKM-variety, meaning a skeletal $B$2-action with $B$3, then $B$4 is encoded by a graph $B$5 whose vertices are the fixed points $B$6, whose edges correspond to one-dimensional $B$7-orbits, and whose edge labels are the corresponding tangential weights. The localization theorem gives

$B$8

Thus equivariant classes become piecewise-polynomial functions satisfying edge-divisibility conditions (Lanini et al., 2021).

If a finite group $B$9 acts on $C$0, normalizing the $C$1-action and permuting fixed points and edge labels, then $C$2 acts on $C$3 by

$C$4

where $C$5 on the right permutes polynomial variables according to its action on $C$6. In type $C$7 this specializes to

$C$8

In this formulation, the dot action is not an auxiliary representation-theoretic decoration; it is built into the automorphism theory of the moment graph itself (Lanini et al., 2021).

A common simplification is to regard the dot action as only a permutation of vertices. The GKM formula shows that this is incomplete: the polynomial variables are permuted simultaneously, and preservation of the divisibility conditions depends on both operations.

3. Flag varieties and regular semisimple Hessenberg varieties

For the full flag variety, the moment graph is the Bruhat graph on $C$9, with edges $A$0 labeled by $A$1. There is an $A$2-basis of $A$3 given by the equivariant Schubert classes $A$4, characterized by support and leading-term divisibility. Under the dot action,

$A$5

so that

$A$6

as an $A$7-module, a direct sum of twisted trivial representations. Passing to ordinary cohomology yields

$A$8

as a permutation representation of $A$9 (Lanini et al., 2021).

For a Hessenberg function $\mathcal{F}\ell_n = GL_n/B$0 and a regular semisimple diagonal matrix $\mathcal{F}\ell_n = GL_n/B$1 with distinct eigenvalues, the regular semisimple Hessenberg variety is

$\mathcal{F}\ell_n = GL_n/B$2

It is $\mathcal{F}\ell_n = GL_n/B$3-stable and satisfies the GKM hypotheses. Its fixed points are indexed by $\mathcal{F}\ell_n = GL_n/B$4, and if $\mathcal{F}\ell_n = GL_n/B$5 is an edge in the moment graph, then the label is

$\mathcal{F}\ell_n = GL_n/B$6

A tuple $\mathcal{F}\ell_n = GL_n/B$7 represents an equivariant class exactly when

$\mathcal{F}\ell_n = GL_n/B$8

for every edge. The equivariant cohomology $\mathcal{F}\ell_n = GL_n/B$9 is a free $T \cong (\mathbb{C}^*)^n$0-module of rank $T \cong (\mathbb{C}^*)^n$1, and ordinary cohomology is the quotient by the ideal $T \cong (\mathbb{C}^*)^n$2 (Brosnan et al., 2015).

On this moment-graph model, $T \cong (\mathbb{C}^*)^n$3 acts by

$T \cong (\mathbb{C}^*)^n$4

and for a simple transposition $T \cong (\mathbb{C}^*)^n$5,

$T \cong (\mathbb{C}^*)^n$6

Setting all $T \cong (\mathbb{C}^*)^n$7 yields the induced dot action on $T \cong (\mathbb{C}^*)^n$8 (Brosnan et al., 2015).

4. Monodromy, characters, and divided differences

In regular semisimple Hessenberg geometry, the dot action has a second realization by monodromy. For a connected reductive complex algebraic group $T \cong (\mathbb{C}^*)^n$9, a Hessenberg subspace $$H_T^*(\mathrm{pt}) = \mathrm{Sym}(X^*(T)\otimes \mathbb{Q}) \cong \mathbb{Q}[x_1,\dots,x_n],$$0 with $$H_T^*(\mathrm{pt}) = \mathrm{Sym}(X^*(T)\otimes \mathbb{Q}) \cong \mathbb{Q}[x_1,\dots,x_n],$$1, and $$H_T^*(\mathrm{pt}) = \mathrm{Sym}(X^*(T)\otimes \mathbb{Q}) \cong \mathbb{Q}[x_1,\dots,x_n],$$2 regular semisimple, one considers

$$H_T^*(\mathrm{pt}) = \mathrm{Sym}(X^*(T)\otimes \mathbb{Q}) \cong \mathbb{Q}[x_1,\dots,x_n],$$3

Over the open set $$H_T^*(\mathrm{pt}) = \mathrm{Sym}(X^*(T)\otimes \mathbb{Q}) \cong \mathbb{Q}[x_1,\dots,x_n],$$4 of regular semisimple elements, the proper map

$$H_T^*(\mathrm{pt}) = \mathrm{Sym}(X^*(T)\otimes \mathbb{Q}) \cong \mathbb{Q}[x_1,\dots,x_n],$$5

produces a local system whose monodromy action of $$H_T^*(\mathrm{pt}) = \mathrm{Sym}(X^*(T)\otimes \mathbb{Q}) \cong \mathbb{Q}[x_1,\dots,x_n],$$6 on $$H_T^*(\mathrm{pt}) = \mathrm{Sym}(X^*(T)\otimes \mathbb{Q}) \cong \mathbb{Q}[x_1,\dots,x_n],$$7 factors through the Weyl group $$H_T^*(\mathrm{pt}) = \mathrm{Sym}(X^*(T)\otimes \mathbb{Q}) \cong \mathbb{Q}[x_1,\dots,x_n],$$8 and coincides with Tymoczko’s dot action. The induced action is independent of the choice of $$H_T^*(\mathrm{pt}) = \mathrm{Sym}(X^*(T)\otimes \mathbb{Q}) \cong \mathbb{Q}[x_1,\dots,x_n],$$9 (Xue, 2020).

For regular semisimple Hessenberg varieties in type $x_i$0, the graded characters of the dot action are controlled by chromatic quasisymmetric functions. Writing $x_i$1 for the character of $x_i$2 on $x_i$3, the main identity is

$x_i$4

equivalently

$x_i$5

where $x_i$6 is the incomparability graph of the associated natural unit interval order (Brosnan et al., 2015). In the equivariant setting, one also has

$x_i$7

for the indifference graph $x_i$8 attached to $x_i$9 (Guay-Paquet, 8 Jul 2025).

A later refinement introduces two commuting $i$0-actions on the ambient ring $i$1: the star action on the right and the dot action on the left. The dot action is

$i$2

Associated Demazure-type operators

$i$3

are dot-equivariant and satisfy $i$4, the braid relations, and a skew-Leibniz rule. Iterating the resulting decomposition theorems gives a direct-sum decomposition of $i$5 into dot-stable summands, categorifying the modular relation for chromatic quasisymmetric functions (Guay-Paquet, 8 Jul 2025).

The geometric constraints on the dot action are correspondingly strong. Hard Lefschetz compatibility implies that cup-product by a $i$6-invariant hyperplane class yields $i$7-equivariant isomorphisms between opposite degrees; in degree $i$8 the dot action is always a permutation representation, and if the Hessenberg variety is connected then $i$9 (Xue, 2020).

5. Extensions beyond the classical Hessenberg setting

One extension replaces the flag variety by quiver Grassmannians for nilpotent representations of the equioriented cycle. Let $A$00 be the cyclic quiver with all arrows oriented in one direction mod $A$01, let $A$02 be a finite-dimensional nilpotent representation, and let

$A$03

A suitable aligned, attractive grading on the coefficient quiver yields a torus $A$04 acting skeletally; fixed points are coordinate subrepresentations parametrized by successor-closed subquivers, one-dimensional orbits correspond to fundamental mutations, and each edge is labeled by

$A$05

This makes $A$06 into a BB-filterable GKM-variety (Lanini et al., 2021).

If certain isomorphism classes of indecomposable summands of $A$07 occur with multiplicities $A$08, then $A$09 embeds into the automorphism group of $A$10 by permuting these isomorphic summands. The resulting action normalizes the torus action, permutes fixed points and moment-graph edges, and acts on equivariant cohomology by

$A$11

with $A$12 permuting the coordinates $A$13 in the weight lattice. Under an additional homogeneity hypothesis on $A$14, the moment graph admits a Palais–Smale orientation and a unique Knutson–Tao basis $A$15 satisfying

$A$16

and

$A$17

as permutation representations of $A$18 (Lanini et al., 2021).

The quiver-Grassmannian case also shows that the action on a natural basis need not be literally by permutation on basis vectors. In the example $A$19 for $A$20, one computes

$A$21

while $A$22 for $A$23, and nevertheless the module still decomposes as a direct sum of twisted trivial summands (Lanini et al., 2021).

A second extension appears in types $A$24 and $A$25, where the Weyl group $A$26 is the group of signed permutations and the relevant object is a spline module $A$27 on an edge-labeled graph $A$28. A spline is a map $A$29 such that along every edge $A$30,

$A$31

The dot action is

$A$32

with $A$33 acting on $A$34 by the standard permutation action on variables. The degree-one piece $A$35 admits explicit generators in four families—$A$36-splines, $A$37-splines, $A$38-splines, and the exceptional $A$39-spline—and the associated left and right dot-action representations display phenomena absent in type $A$40, including the character $A$41 and certain double-interval blocks in the $A$42-splines (Lesnevich, 17 Nov 2025).

6. General reductive type and the $A$43 classification

In arbitrary reductive type, the monodromic definition and the geometric constraints described above permit explicit classification in small rank. In type $A$44, there are exactly eight Hessenberg ideals $A$45, and for each one the paper computes the Poincaré-character polynomial

$A$46

where $A$47 is the character of the Weyl-group representation on $A$48 (Xue, 2020).

Hessenberg ideal $A$49	$A$50
$A$51	$A$52
$A$53	$A$54
$A$55	$A$56
$A$57	$A$58
$A$59	$A$60
$A$61	$A$62
$A$63	$A$64
$A$65	$A$66

These formulas show that the dot action in $A$67 is not exhausted by inductions from trivial or sign characters on standard Levi subgroups. In particular, $A$68 carries a factor $A$69, which is neither trivial on any proper $A$70 nor a sign twist thereof. The paper therefore concludes that the naive generalization of the Stanley–Stembridge positivity conjecture—namely that every $A$71 is a sum of $A$72—fails already in rank $A$73 (Xue, 2020).

Taken together, these results place Tymoczko’s dot action at the intersection of GKM theory, monodromy, Schubert-calculus-type bases, and combinatorial representation theory. In type $A$74 it yields permutation representations linked to chromatic quasisymmetric functions; in quiver Grassmannians it produces new permutation-group actions on cohomology; and in other Lie types it supports genuinely non-type-$A$75 phenomena, both algebraically and geometrically.