---
title: Tymoczko's Dot Action in Equivariant Cohomology
url: https://www.emergentmind.com/topics/tymoczko-s-dot-action
type: topic
---

# Tymoczko's Dot Action in Equivariant Cohomology

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 [2105.10122] [1511.00773] [2007.08712] [2511.13913].

## 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
\[
H_T^*(\mathcal{F}\ell_n)\hookrightarrow \bigoplus_{w\in S_n}\mathbb{Q}[x_1,\dots,x_n],
\]
sending a class \(\alpha\) to its collection of fixed-point values \((\alpha|_w)_w\) [2105.10122].

On polynomials, the dot action of \(w\in S_n\) is
\[
(w\cdot f)(x_1,\dots,x_n)=f(x_{w(1)},\dots,x_{w(n)}).
\]
Equivalently, \(w\) permutes the weights, or Chern roots, on \(H_T^*(\mathrm{pt})\). If \(\alpha\in H_T^*(\mathcal{F}\ell_n)\) is represented by its localizations \((\alpha|_u)_{u\in S_n}\), then Tymoczko’s dot action is defined by
\[
(w\cdot \alpha)|_u = w\bigl(\alpha|_{w^{-1}u}\bigr),
\]
where \(w\) on the right acts on the polynomial by permuting variables [2105.10122].

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 \(X\) is a GKM-variety, meaning a skeletal \(T\)-action with \(H^{\mathrm{odd}}(X)=0\), then \(H_T^*(X)\) is encoded by a graph \(G\) whose vertices are the fixed points \(X^T\), whose edges correspond to one-dimensional \(T\)-orbits, and whose edge labels are the corresponding tangential weights. The localization theorem gives
\[
H_T^*(X)\cong \left\{\,f:G_0\to \mathbb{Q}[x_1,\dots,x_r]\;\middle|\; f(x)-f(y)\ \text{divisible by }\alpha_E\ \text{for each edge }E:x\to y\,\right\}.
\]
Thus equivariant classes become piecewise-polynomial functions satisfying edge-divisibility conditions [2105.10122].

If a finite group \(W\) acts on \(X\), normalizing the \(T\)-action and permuting fixed points and edge labels, then \(W\) acts on \(H_T^*(X)\) by
\[
(w\cdot f)(x)=w\bigl(f(w^{-1}\cdot x)\bigr),
\]
where \(w\) on the right permutes polynomial variables according to its action on \(X^*(T)\). In type \(A\) this specializes to
\[
(w\cdot f)(u)=f(w^{-1}u)(x_{w(1)},\dots,x_{w(n)}).
\]
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 [2105.10122].

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 \(S_n\), with edges \(u\to us_{i,i+1}\) labeled by \(x_{u(i)}-x_{u(i+1)}\). There is an \(R\)-basis of \(H_T^*(\mathcal{F}\ell_n)\) given by the equivariant Schubert classes \(\sigma_w\), characterized by support and leading-term divisibility. Under the dot action,
\[
w\cdot \sigma_u=\sigma_{wu},
\]
so that
\[
H_T^*(\mathcal{F}\ell_n)\cong \bigoplus_{u\in S_n} R(-2\ell(u))
\]
as an \(S_n\)-module, a direct sum of twisted trivial representations. Passing to ordinary cohomology yields
\[
H^*(\mathcal{F}\ell_n)\cong \bigoplus_{u\in S_n}\mathbb{C}(-2\ell(u))
\]
as a permutation representation of \(S_n\) [2105.10122].

For a Hessenberg function \(h\) and a regular semisimple diagonal matrix \(X=\mathrm{diag}(\lambda_1,\dots,\lambda_n)\) with distinct eigenvalues, the regular semisimple Hessenberg variety is
\[
\mathrm{Hess}(h)=\{F\in \mathcal{F}\ell_n : X\cdot F_i\subseteq F_{h(i)}\}.
\]
It is \(T\)-stable and satisfies the GKM hypotheses. Its fixed points are indexed by \(w\in S_n\), and if \(e=(w\to w\cdot s_i)\) is an edge in the moment graph, then the label is
\[
\alpha_e=t_{w(i)}-t_{w(i+1)}.
\]
A tuple \((f_w)_w\in \prod_w \mathbb{Q}[t_1,\dots,t_n]\) represents an equivariant class exactly when
\[
f_w-f_{w\cdot s_i}\equiv 0 \pmod{t_{w(i)}-t_{w(i+1)}}
\]
for every edge. The equivariant cohomology \(H_T^*(\mathrm{Hess}(h))\) is a free \(\mathbb{Q}[t_1,\dots,t_n]\)-module of rank \(n!\), and ordinary cohomology is the quotient by the ideal \((t_1,\dots,t_n)\) [1511.00773].

On this moment-graph model, \(S_n\) acts by
\[
(\sigma\cdot f)_w=\sigma\bigl(f_{\sigma^{-1}w}\bigr),
\]
and for a simple transposition \(s_i\),
\[
(s_i\cdot f)_w=s_i(f_{s_i\cdot w}).
\]
Setting all \(t_i=0\) yields the induced dot action on \(H^*(\mathrm{Hess}(h))\) [1511.00773].

## 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 \(G\), a Hessenberg subspace \(M\subset \mathfrak{g}\) with \(\mathfrak{b}\subset M\), and \(y\in \mathfrak{g}\) regular semisimple, one considers
\[
\mathrm{Hess}(M,y)=\{gB\in G/B\mid g^{-1}\cdot y\in M\}.
\]
Over the open set \(\mathfrak{g}^{rs}\) of regular semisimple elements, the proper map
\[
\pi_M:G\times^B M\to \mathfrak{g},\qquad \pi_M(g,x)=g\cdot x,
\]
produces a local system whose monodromy action of \(\pi_1(\mathfrak{g}^{rs},y)\) on \(H^*(\mathrm{Hess}(M,y))\) factors through the Weyl group \(W\) and coincides with Tymoczko’s dot action. The induced action is independent of the choice of \(y\in\mathfrak{g}^{rs}\) [2007.08712].

For regular semisimple Hessenberg varieties in type \(A\), the graded characters of the dot action are controlled by chromatic quasisymmetric functions. Writing \(\chi_d\) for the character of \(S_n\) on \(H^{2d}(\mathrm{Hess}(h))\), the main identity is
\[
\sum_{d=0}^{|h|-1} \mathrm{ch}(\chi_d)\, t^d = \omega X_G(t),
\]
equivalently
\[
\mathrm{ch}(\chi_d)=[t^d]\omega X_G(t),
\]
where \(G\) is the incomparability graph of the associated natural unit interval order [1511.00773]. In the equivariant setting, one also has
\[
\mathrm{Frob}_q\bigl(H_T^*(\mathrm{Hess}(S,h))\bigr)=X_{G(h)}(q),
\]
for the indifference graph \(G(h)\) attached to \(h\) [2507.05614].

A later refinement introduces two commuting \(S_n\)-actions on the ambient ring \(H=\mathrm{Fun}(S_n,\mathbb{C}[t_1,\dots,t_n])\): the star action on the right and the dot action on the left. The dot action is
\[
(w\cdot f)(v;t_1,\dots,t_n)=f(w^{-1}v;t_{w(1)},\dots,t_{w(n)}),
\qquad
w\cdot t_i=t_{w(i)},\quad w\cdot x_i=x_i.
\]
Associated Demazure-type operators
\[
\partial_i(f)=\frac{f*(1-s_i)}{x_i-x_{i+1}}
\]
are dot-equivariant and satisfy \(\partial_i^2=0\), the braid relations, and a skew-Leibniz rule. Iterating the resulting decomposition theorems gives a direct-sum decomposition of \(H_{C(h)}\) into dot-stable summands, categorifying the modular relation for chromatic quasisymmetric functions [2507.05614].

The geometric constraints on the dot action are correspondingly strong. Hard Lefschetz compatibility implies that cup-product by a \(W\)-invariant hyperplane class yields \(W\)-equivariant isomorphisms between opposite degrees; in degree \(0\) the dot action is always a permutation representation, and if the Hessenberg variety is connected then \(H^0\cong \mathbf{1}\) [2007.08712].

## 5. Extensions beyond the classical Hessenberg setting

One extension replaces the flag variety by quiver Grassmannians for nilpotent representations of the equioriented cycle. Let \(Q=A_n\) be the cyclic quiver with all arrows oriented in one direction mod \(n\), let \(M\) be a finite-dimensional nilpotent representation, and let
\[
\mathrm{Gr}_e(M)=\{\,U\subset M\mid U\ \text{a subrepresentation with }\dim U=e\,\}.
\]
A suitable aligned, attractive grading on the coefficient quiver yields a torus \(T=(\mathbb{C}^*)^{d+1}\) 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
\[
\alpha_p=E_j-E_i+(k'-k)\varepsilon.
\]
This makes \((\mathrm{Gr}_e(M),T)\) into a BB-filterable GKM-variety [2105.10122].

If certain isomorphism classes of indecomposable summands of \(M\) occur with multiplicities \(k_1,\dots,k_m\), then \(G_{k_1}\times\cdots\times G_{k_m}\) embeds into the automorphism group of \(M\) 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
\[
(\sigma\cdot f)(S)=\sigma\bigl(f(\sigma^{-1}\cdot S)\bigr),
\]
with \(\sigma\) permuting the coordinates \(E_i\) in the weight lattice. Under an additional homogeneity hypothesis on \(M\), the moment graph admits a Palais–Smale orientation and a unique Knutson–Tao basis \(\{p_S\}_{S\in X^T}\) satisfying
\[
\sigma\cdot p_S=p_{\sigma S},
\]
and
\[
H_T^*(X)\cong \bigoplus_{S\in X^T} R(-2\dim S),
\qquad
H^*(X)\cong \bigoplus_{S\in X^T}\mathbb{C}(-2\dim S),
\]
as permutation representations of \(G_k\) [2105.10122].

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 \(X=\mathrm{Gr}_4(M)\) for \(M=A_4\oplus A_2\oplus A_2\), one computes
\[
s\cdot p_2=p_2+(E_2-E_1)p_1,
\]
while \(s\cdot p_i=p_i\) for \(i\neq 2\), and nevertheless the module still decomposes as a direct sum of twisted trivial summands [2105.10122].

A second extension appears in types \(B\) and \(C\), where the Weyl group \(W=B_n=C_n\) is the group of signed permutations and the relevant object is a spline module \(\mathcal{M}_H\) on an edge-labeled graph \(G_H\). A spline is a map \(\rho:W\to \mathbb{C}[x_1,\dots,x_n]\) such that along every edge \((w,w\cdot s_\alpha)\),
\[
\rho(w)-\rho(w\cdot s_\alpha)\in \langle w(\alpha)\rangle.
\]
The dot action is
\[
(w\cdot \rho)(v):=w[\rho(w^{-1}v)],
\]
with \(w\) acting on \(\mathbb{C}[x]\) by the standard permutation action on variables. The degree-one piece \(\mathcal{M}_H^1\) admits explicit generators in four families—\(f\)-splines, \(y\)-splines, \(g\)-splines, and the exceptional \(h\)-spline—and the associated left and right dot-action representations display phenomena absent in type \(A\), including the character \(\delta:w\mapsto (-1)^{\mathrm{neg}(w)}\) and certain double-interval blocks in the \(y\)-splines [2511.13913].

## 6. General reductive type and the \(G_2\) classification

In arbitrary reductive type, the monodromic definition and the geometric constraints described above permit explicit classification in small rank. In type \(G_2\), there are exactly eight Hessenberg ideals \(I\), and for each one the paper computes the Poincaré-character polynomial
\[
P_M(q)=\sum_{i=0}^n \chi_{H^{2i}(\mathrm{Hess}(M,y))}\, q^i,
\]
where \(\chi_{H^{2i}(\cdots)}\) is the character of the Weyl-group representation on \(H^{2i}\) [2007.08712].

| Hessenberg ideal \(I\) | \(P_M(q)\) |
|---|---|
| \(I_{\emptyset}\) | \(1 + 2q + 2q^2 + 2q^3 + 2q^4 + 2q^5 + q^6\) |
| \(I_{2\beta+3\alpha}\) | \(1 + 2q + (2+\varepsilon_1)q^2 + (2+\varepsilon_1)q^3 + 2q^4 + q^5\) |
| \(I_{\beta+3\alpha}\) | \(1 + (2+\varepsilon_1)q + (2+2\varepsilon_1)q^2 + (2+\varepsilon_1)q^3 + q^4\) |
| \(I_{\beta+2\alpha}\) | \(1 + (2+\varepsilon_1+\chi_2)q + (2+\varepsilon_1+\chi_2)q^2 + q^3\) |
| \(I_{\beta+\alpha}\) | \(1 + (2+\varepsilon_1+2\chi_2+\chi_1+\varepsilon_2)q + q^2\) |
| \(I_{\alpha}\) | \((1 + \chi_2 + \chi_1 + \varepsilon_2)(1+q)\) |
| \(I_{\beta}\) | \((1 + \varepsilon_1 + \chi_2 + \chi_1)(1+q)\) |
| \(I_{\alpha,\beta}(=\mathfrak{u})\) | \(1 + \varepsilon_1 + 2\chi_2 + 2\chi_1 + \varepsilon_2\) |

These formulas show that the dot action in \(G_2\) is not exhausted by inductions from trivial or sign characters on standard Levi subgroups. In particular, \(I_{\beta+\alpha}\) carries a factor \(\chi_1+\varepsilon_2\), which is neither trivial on any proper \(W_J\) nor a sign twist thereof. The paper therefore concludes that the naive generalization of the Stanley–Stembridge positivity conjecture—namely that every \(P_M(1)\) is a sum of \(\mathrm{Ind}_{W_J}^W 1\)—fails already in rank \(2\) [2007.08712].

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 \(A\) 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-\(A\) phenomena, both algebraically and geometrically.

Source: https://www.emergentmind.com/topics/tymoczko-s-dot-action