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#The dot action

The dot action is a representation of \(\symS_n\) on the cohomology of certain Hessenberg varieties. It was introduced by J. Tymoczko in the setting of equivariant cohomology [Tym08, Tym08]. For regular semisimple Hessenberg varieties, this action gives the representation-theoretic side of the Shareshian–Wachs conjecture. The conjecture was proved by P. Brosnan and T. Chow [BC18], and independently by M. Guay-Paquet [Gua16].

The broader chromatic symmetric function background is discussed on the chromatic quasisymmetric function page. Here we isolate the representation-theoretic mechanism: start with a Hessenberg sequence, build the GKM (M. Goresky, R. Kottwitz, and R. MacPherson) model for equivariant cohomology, descend the dot action to ordinary cohomology, and finally convert the resulting graded characters into a Frobenius characteristic.

#Hessenberg sequences and varieties

A Hessenberg sequence is a weakly increasing sequence \(m=(m_1,\dotsc,m_n)\) such that \[i\leq m_i\leq n \qquad\text{for all } i.\] Equivalently, it is a Hessenberg function \(m:[n]\to[n]\) with \(m(i)\geq i.\) This is the same data that appears in the area-sequence description of unit interval graphs, although the indexing convention depends on whether one starts from the Hessenberg function or from the Dyck-diagram picture. In the diagram convention used on the chromatic quasisymmetric function page, the Hessenberg sequence records the number of non-square boxes in each row of the extended \(n\times n\) diagram; for instance the area sequence \((0,1,2,1,1)\) corresponds to \(m=(2,3,5,5,5).\) We write \(\avec\) for the corresponding area sequence and \(\Gamma_\avec\) for the associated graph.

Let \(F(n)\) be the set of all flags \[F_1\subset F_2\subset \dotsb \subset F_n=\setC^n, \qquad \dim F_i=i.\] Let \(D\) be a diagonal \(n\times n\) matrix with distinct diagonal entries. The type \(A\) regular semisimple Hessenberg variety associated with \(m\) is \[H(m) \coloneqq \{F\in F(n):D F_i\subseteq F_{m_i}\text{ for all }i\}.\] For background on this construction, see the thesis of N. Teff [Tef13] and J. Tymoczko’s introduction to equivariant cohomology and homology [Tym05].

The same sequence also determines a natural unit interval graph \(\Gamma_\avec\) on vertex set \([n].\) In the standard Hessenberg-function convention, we place an edge \(\{i,j\},\) with \(i \lt j,\) whenever \[j\leq m_i.\] For this graph, the Shareshian–Wachs chromatic quasisymmetric function is \[\chrom_{\Gamma_\avec}(\xvec;q) = \sum_{\substack{\kappa:[n]\to\setP\\ \kappa \text{ proper}}} x_{\kappa(1)}\dotsm x_{\kappa(n)}q^{\asc(\kappa)},\] where the orientation is the increasing orientation \(i\to j\) for \(i \lt j.\)

The original Shareshian–Wachs conjecture [Conj. 5.3, SW12] predicts that the cohomology of \(H(m)\) carries a graded \(\symS_n\)-action whose Frobenius characteristic is \(\omega\chrom_{\Gamma_\avec}(\xvec;q).\) This was proved by P. Brosnan and T. Chow [BC18], and independently by M. Guay-Paquet [Gua16]. There is also a closely related regular nilpotent story involving acyclic orientations and Tymoczko cells; see [NT24].

#The GKM model

Let \(T=(\setC^\ast)^n\) be the diagonal torus. We write \[H_T^\ast(\mathrm{pt})=\setC[t_1,\dotsc,t_n].\] The \(T\)-fixed points of \(H(m)\) are naturally indexed by permutations \(w\in\symS_n.\) The moment graph records the zero- and one-dimensional \(T\)-orbits in \(H(m).\) The corresponding moment graph, denoted \(MG(\avec)\) on the chromatic quasisymmetric function page, has these permutations as vertices. In the right-multiplication convention used below, the edges are \[w\longleftrightarrow w s_{ij} \qquad \text{for }\{i,j\}\in E(\Gamma_\avec).\]

The GKM description realizes the equivariant cohomology as a ring of tuples \[p=(p_w)_{w\in\symS_n}, \qquad p_w\in \setC[t_1,\dotsc,t_n],\] satisfying one divisibility condition for each edge of the moment graph. For every edge \(\{i,j\}\) of \(\Gamma_\avec\) and every \(w\in\symS_n,\) we require \[p_w-p_{w s_{ij}} \quad\text{is divisible by}\quad t_{w(i)}-t_{w(j)}.\] Here \(s_{ij}\) is the transposition swapping \(i\) and \(j.\) Thus \[H_T^\ast(H(m)) = \left\{ (p_w)_{w\in\symS_n} : p_w-p_{w s_{ij}}\in \langle t_{w(i)}-t_{w(j)} \rangle \text{ for all } w \text{ and } \{i,j\}\in E(\Gamma_\avec) \right\}.\]

The ordinary cohomology is obtained by setting the equivariant parameters to zero: \[H^\ast(H(m)) \cong H_T^\ast(H(m)) \otimes_{\setC[t_1,\dotsc,t_n]}\setC,\] where \(t_1,\dotsc,t_n\) act as zero on \(\setC.\) Equivalently, we quotient by the ideal generated by the positive-degree polynomial variables.

#Definition of the dot action

The group \(\symS_n\) acts on \(\setC[t_1,\dotsc,t_n]\) by permuting the indices: \(\sigma(t_i)\coloneqq t_{\sigma(i)}.\) Tymoczko’s dot action on a GKM tuple is \[(\sigma\cdot p)_w \coloneqq \sigma\left(p_{\sigma^{-1}w}\right).\] The subscript \(\sigma^{-1}w\) moves the fixed-point label, and the outer \(\sigma\) permutes the polynomial variables. The GKM divisibility relations are preserved, so this gives an action on equivariant cohomology. Since the ideal \((t_1,\dotsc,t_n)\) is \(\symS_n\)-stable, the action descends to ordinary cohomology.

The grading convention used in the chromatic formula is that polynomial degree \(d\) corresponds to cohomological degree \(2d.\)

#From the action to a symmetric function

We now describe the steps needed to do computations with Tymoczko’s dot action.

  1. Fix a Hessenberg sequence \(m,\) form the corresponding area sequence \(\avec,\) and form the graph \(\Gamma_\avec.\)

  2. For each degree \(d,\) consider the vector space of all assignments \[\symS_n \to \setC[t_1,\dotsc,t_n]_d.\] This vector space has dimension \[n!\binom{n+d-1}{n-1},\] since each of the \(n!\) tuple entries carries an independent copy of \(\setC[t_1,\dotsc,t_n]_d.\)

  3. Translate each GKM divisibility relation into linear equations in the coefficients of the homogeneous polynomials; the edges of \(\Gamma_\avec\) determine which equations occur. The solution space \(M_d\) is the degree \(d\) part of the equivariant GKM module.

  4. To pass to ordinary cohomology in degree \(d,\) quotient \(M_d\) by \[R_d=\sum_{i=1}^n t_i M_{d-1}.\] Equivalently, \(R_d\) is the degree \(d\) part of \((t_1,\dotsc,t_n)H_T^\ast(H(m)).\)

  5. For each partition \(\mu\vdash n,\) choose a representative \(\sigma_\mu\) of the corresponding conjugacy class and compute the trace \[\chi_d(\mu) = \mathrm{tr}\left( \sigma_\mu: H^{2d}(H(m))\to H^{2d}(H(m)) \right).\] Since characters are class functions, the trace only needs to be computed on one representative of each conjugacy class. In practice, choose a basis for the quotient \(M_d/R_d,\) compute the matrix of \(\sigma_\mu\) on this quotient, and take its trace. The trace is independent of the chosen quotient basis.

  6. These trace values determine the Frobenius characteristic: \[\frobChar_d = \sum_{\mu\vdash n}\chi_d(\mu)\frac{\powerSum_\mu}{z_\mu}.\] The full graded Frobenius characteristic is \[\sum_d q^d \frobChar_d.\]

#The Shareshian–Wachs theorem

The main theorem connecting the above representation to symmetric functions is the following.

Theorem (Brosnan–Chow, Guay-Paquet).

Let \(m\) be a Hessenberg sequence, and let \(\Gamma_\avec\) be the corresponding natural unit interval graph. Then \[\sum_{d\geq 0} q^d \frobChar\left(H^{2d}(H(m))\right) = \omega \chrom_{\Gamma_\avec}(\xvec;q).\]

The right-hand side is the chromatic quasisymmetric function with the \(\omega\) involution applied. Consequently, the theorem implies Schur positivity of \(\omega\chrom_{\Gamma_\avec}(\xvec;q).\)

M. Guay-Paquet gives a representation-level explanation of the modular relation using divided difference operators on the GKM model [Gua25]. In the stable and almost-stable cases these operators decompose the equivariant cohomology module into dot-action subrepresentations, categorifying the modular relation satisfied by chromatic quasisymmetric functions [Thm. 5.1 and Thm. 5.3, Gua25].

T. Horiguchi, M. Masuda, and T. Sato give elementary GKM-theoretic proofs of the modular law for the geometric objects behind the Shareshian–Wachs and unicellular LLT correspondences [HMS23]. This includes the regular semisimple Hessenberg varieties and the twin varieties appearing in the LLT setting.

S. Kato gives a different geometric realization of the chromatic symmetric function of a unit interval graph using a smooth projective variety \(X_\Gamma\) and a map to the affine Grassmannian [Kat24]. The decomposition theorem and geometric Satake identify graded multiplicity spaces whose Schur-character sum is the chromatic symmetric function [Thm. A, Kat24]. Kato also formulates a geometric Stanley–Stembridge conjecture which would imply the Shareshian–Wachs conjecture for these varieties.

K. Salois studies the regular semisimple Hessenberg varieties with \(h=(h(1),n,\dotsc,n)\) and the transpose family \(h'=((n-1)^{n-m},n^m)\) [Sal26]. For these families he constructs higher Specht bases and permutation bases for the cohomology rings, together with bijections to \(P_h\)-tableaux. The permutation-basis result gives a geometric explanation of the corresponding elementary-positivity statement.

#Types B and C

N. R. T. Lesnevich extends the same GKM/spline viewpoint to types \(B\) and \(C\) [Les25]. The vertices are the signed permutations \(\symB_n,\) and an order ideal \(H\) in the positive roots determines the signed transpositions and edge relations. The resulting spline module \(\mathcal M_H\) carries a dot action and has two natural quotients: a left quotient giving the ordinary-cohomology representation for regular semisimple Hessenberg varieties in types \(B\) and \(C,\) and a right quotient related to the corresponding isospectral-matrix manifolds and unicellular LLT polynomials. Lesnevich computes the degree-one pieces of both dot-action representations for all such \(H\) directly from the combinatorics of these signed transpositions.

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