Geometry of the twin manifolds of regular semisimple Hessenberg varieties and unicellular LLT polynomials

Recently, Masuda-Sato and Precup-Sommers independently proved an LLT version of the Shareshian-Wachs conjecture which says that the Frobenius characteristics of the cohomology of the twin manifolds of regular semisimple Hessenberg varieties are unicellular LLT polynomials. The purpose of this paper is to study the geometry of twin manifolds and we prove that they are related by explicit blowups and fiber bundle maps. Upon taking their cohomology, we obtain a direct proof of the modular law which establishes the LLT Shareshian-Wachs conjecture.


Introduction
LLT polynomials are symmetric functions that serve as q-deformations of the product of Schur functions, introduced by Lascoux, Leclerc, and Thibon [18] in their study of quantum affine algebras.A specific class of these polynomials known as unicellular LLT polynomials (Definition 2.2) was explored in [9] using Dyck paths, or Hessenberg functions, in parallel with the chromatic quasisymmetric functions.The purpose of this paper is to investigate on the geometry of the twin manifolds of regular semisimple Hessenberg varieties (Propositions 4.6 and 4.7) and provide a direct geometric proof of the fact that unicellular LLT polynomials are the Frobenius characteristics of representations of symmetric groups S n on the cohomology of the twin manifolds (Theorem 5.4).
Hessenberg varieties are subvarieties of flag varieties with interesting properties in geometric, representation theoretic and combinatorial aspects (cf.[11,1]).One of their notable features is the S n -action on their cohomology [27], where the induced graded S n -representations are equivalent to the purely combinatorially defined symmetric functions known as the chromatic quasisymmetric functions [24,23] of the associated indifference graphs.This equivalence (cf.(5.5)), known as the Shareshian-Wachs conjecture [23], proved in [8,14], translates the longstanding conjecture by Stanley and Stembridge [25] on e-positivity of the chromatic (quasi)symmetric functions into a positivity problem on the S n -representations on the cohomology of Hessenberg varieties.
A natural question arises whether there exist geometric objects that encode unicellular LLT polynomials through their cohomology, as in the Shareshian-Wachs conjecture.Recently, an answer was found by Masuda-Sato and Precup-Sommers in [21,22] where they proved that unicellular LLT polynomials are the Frobenius characteristics of the cohomology of the twin manifolds of regular semisimple Hessenberg varieties.
The unitary group U (n) is acted on by its maximal torus T = U (1) n by left and right multiplications.So we have the quotient maps where X denotes the flag variety Fl(n) and Y denotes the isospectral manifold of Hermitian matrices with a fixed spectrum (cf.§3.1).The twin manifold of a Hessenberg variety X h ⊂ Fl(n) = X is now defined in [6] as the submanifold Y h := p 1 (p −1 2 (X h )) of the isospectral manifold Y .These twin manifolds Y h , which are the spaces of staircase Hermitian matrices with a fixed given spectrum, are interesting compact orientable smooth real algebraic varieties.They generalize the space of tridiagonal matrices of a given spectrum [26,7,10] and we have natural isomorphisms (1.1) ) which induce an S n -action on the cohomology H * (Y h ) from that on H * (X h ) in [27].The LLT analogue of the Shareshian-Wachs conjecture (LLT-SW conjecture, for short) tells us that the Frobenius characteristics of H * (Y h ) are the unicellular LLT polynomials (cf.Theorem 5.4).The known proofs in [21,22] are rather indirect and use only the Hessenberg varieties without looking into the geometry of twin manifolds themselves.See Remark 5.5 for more details.Therefore, it seems natural to ask for a direct approach through the geometry of twin manifolds.
The modular law (cf.Definition 2.4), introduced in [3,13] for chromatic quasisymmetric functions and in [5,19] for unicellular LLT polynomials, is a significant relation involving specific triples of these functions.It serves as a symmetric function analogue to the well known deletion-contraction relation of chromatic polynomials.In fact, Abreu and Nigro proved in [3] that together with an initial condition (for the case of h(i) = n for all i) and the multiplicativity (cf.(2.5)), the modular law completely determines the chromatic quasisymmetric functions and unicellular LLT polynomials.
In [16], the authors investigated on the geometry of Hessenberg varieties X h and proved that the Hessenberg varieties X h − , X h and X h + for a modular triple h = (h − , h, h + ) (cf.Definition 2.3) are related by explicit blowups and projective bundle maps.By applying the blowup formula and projective bundle formula, we then immediately obtain the modular law for the cohomology of X h , which provides us with an elementary proof of the Shareshian-Wachs conjecture.
In this paper, we investigate on the geometry of the twin manifolds Y h .The key for our comparison of twin manifolds is the roof manifold Y h defined in Definition 4.1.For a modular triple h = (h − , h, h + ) (cf.Definition 2.3), we construct maps where pr 2 is a smooth fibration over the complex projective line P 1 with fiber Y h (Proposition 4.6) and π is the blowup along the submanifold Y h − of complex codimension 2 (Proposition 4.7).We define an S n -action on the cohomology H * ( Y h ) and show that the induced maps on cohomology by π and pr 2 are S n -equivariant.We thus obtain S n -equivariant isomorphisms Upon taking the Frobenius characteristic, we have the modular law for H * (Y h ) and hence the LLT-SW conjecture where LLT h (q) denotes the unicellular LLT polynomial associated to h.
The layout of this paper is as follows.In §2, we review the definition of unicelullar LLT polynomials and their characterization by the modular law.In §3, we review the results in [6] on twin manifolds including the S n -action defined on their cohomology.In §4, we study the geometry of twin manifolds of triples and in §5, we establish the modular law for S n -representations on their cohomology.
All cohomology groups in this paper have rational coefficients.By P r , we denote the complex projective space of one dimensional subspaces in C r+1 .
Acknowledgement.We thank Anton Ayzenberg, Jaehyun Hong, Antonio Nigro and Takashi Sato for enlightening discussions and comments.

Unicellular LLT polynomials
LLT polynomials are symmetric functions introduced by Lascoux, Leclerc, and Thibon [18] as q-deformations of the product of Schur functions in their study of quantum affine algebras.In the case of unicellular LLT polynomials, which form a subfamily of these symmetric functions, a more convenient model is presented in [9] using Hessenberg functions.This model represents unicellular LLT polynomials as symmetric functions that encode colorings of graphs, which may not be proper.
In this section, we recall the definition of unicellular LLT polynomials in terms of Hessenberg functions from [9] and the characterization by the modular law from [4].
Every unicellular LLT polynomial can be written as a symmetric function which encodes vertex-colorings of the indifference graph Γ h , similar to the definition of the chromatic quasisymmetric function [24,23].
A map γ : V(Γ h ) → N is said to be a (vertex-)coloring of Γ h , and it is said to be proper if and only if γ(i) = γ(j) whenever (i, j) ∈ E(Γ h ), where N is the set of colors indexed by positive integers.(1) The unicellular LLT polynomial associated to h is where γ runs over all colorings which are not necessarily proper and (2) The chromatic quasisymmetric function associated to h is where γ runs over all proper colorings.(1) If h(j) = h(j + 1) and h −1 (j) = {j 0 } for some 1 ≤ j 0 < j < n, then h − and h + are defined by (2) If h(j) + 1 = h(j + 1) = j + 1 and h −1 (j) = ∅ for some 1 ≤ j < n, then h − and h + are defined by The two conditions (1) and ( 2) in Definition 2.3 are actually dual to each other.See Remark 4.2.
Definition 2.4 (Modular law).Let F be a function from the set of Hessenberg functions to Λ[q].We say that F satisfies the modular law if for every modular triple (h − , h, h + ).
Note that unicellular LLT polynomials and chromatic quasisymmetric functions can be viewed as functions LLT (−) and csf (−) from the set of Hessenberg functions to Λ[q].Proposition 2.5.[3,5,19] Unicellular LLT polynomials and chromatic quasisymmetric functions satisfy the modular law.
The modular law, analogous to the deletion-contraction property in chromatic polynomials, plays a crucial role in determining these symmetric functions recursively.
Remark 2.7.One fundamental technique in representation theory and combinatorics is to construct a geometric object corresponding to an object of interest.Hard combinatorics problems are often translated into well known geometry problems and solved subsequently, as demonstrated by the recent spectacular works of June Huh.The Shareshian-Wachs conjecture formulated in [23] and proved in [8,14] tells us that the chromatic quasisymmetric function (2.2) is the ω-dual of the Frobenius characteristic of regular semisimple Hessenberg varieties X h in §3.2 below.Here ω is an involution of Λ interchanging each Schur function with its transpose.The first two conditions (1) and (2) in Theorem 2.6 for csf are easy to check for ωF(h) and hence the Shareshian-Wachs conjecture follows immediately from the modular law (2.3) for F(h).In [16], we investigated on the geometry of generalized Hessenberg varieties and constructed canonical S n -equivariant isomorphisms Upon taking the Frobenius characteristic, we obtain the modular law (2.3) and hence the Shareshian-Wachs conjecture In the remainder of this paper, we will prove that the three conditions in Theorem 2.6 for unicellular LLT are satisfied for the Frobenius characteristics of representations of S n on the cohomology of the twin manifolds Y h of regular semisimple Hessenberg varieties X h , by finding geometric relations among the twin manifolds that give rise to the modular law (2.3)upon taking cohomology.This will give us a direct proof of the LLT-SW conjecture (cf.Theorem 5.4) without relying on the Shareshian-Wachs conjecture.

Twin manifolds and their cohomology
In this section, we collect necessary facts about the twin manifolds Y h of regular semisimple Hessenberg varieties X h of type A from [6].
Let h : [n] → [n] be a Hessenberg function (Definition 2.1) where be a fixed regular semisimple diagonal matrix.Let T ∼ = U (1) n denote the group of diagonal unitary matrices.
3.1.Isospectral manifolds.Let H denote the real vector space of n × n Hermitian matrices.Let Y = Y (x) ⊂ H be the set of n × n Hermitian matrices whose characteristic polynomial is n i=1 (t − λ i ).In other words, Y is the set of n × n Hermitian matrices with fixed (unordered) spectrum {λ i }.As the diffeomorphism type of Y is independent of x by [6, Theorem 3.5], we will suppress x to simplify the notation.
By the spectral decomposition theorem in linear algebra, any matrix in Y is of the form g −1 xg, with g ∈ U (n) and the map In particular, Y is a compact smooth orientable manifold of real dimension which is a smooth projective variety of real dimension n 2 −n.As the columns of a unitary matrix define a flag in C n , we have The isospectral manifold Y and the flag variety X fit into the following diagram where p 1 is the left quotient and p 2 is the right quotient.

Hessenberg varieties and their twins. For a Hessenberg function
, the Hessenberg variety associated to h is defined in [11] as Under our assumptions, by [11], the Hessenberg variety X h is a smooth projective variety of real dimension See [1] for a recent survey on Hessenberg varieties.The twin manifold Y h of X h is a submanifold of Y defined in [6] by where p 1 and p 2 are the quotient maps in (3.2).
By [6,Theorem 3.5], Y h is a compact real smooth manifold of dimension (3.3) whose diffeomorphism type is independent of the choice of x.Using (3.1), it is straightforward to check that Y h is precisely, the locus of staircase matrices where y ij = y ji denotes the entry of the Hermitian matrix y at the i-th row and j-th column.
Since the real dimension of Y is of Y h coincides with the actual dimension (3.3).In particular, Y h is the transversal vanishing locus of where (i, j) runs over the pairs with h(i) < j, or equivalently h(j) < i.By (3.1), we have a right action of T on Y by In particular, we have an induced T -action on Y h for every Hessenberg function h, whose fixed point set is exactly Note that our notation is different from that in [6], where Hessenberg varieties and their twin manifolds are denoted by Y h and X h respectively.
3.3.Goresky-Kottwitz-MacPherson theory.When a manifold admits a nice torus action, we can compute its equivariant cohomology from the data of 0 and 1-dimensional orbits.Definition 3.1.(See [6, Definition 5.1].)A compact orientable manifold M with a smooth action of a compact torus T = U (1) n is called a GKM manifold if it satisfies the following conditions.
(2) The set M T of T -fixed points is finite.
(3) The weights of the induced representation of T on the tangent space of M at each y ∈ M T are pairwisely non-collinear: if then α i and α j are non-collinear as vectors whenever i = j.
for some α ∈ Hom(T, U (1)) ∼ = Z n .In particular, each T -invariant 2-sphere connects precisely two T -fixed points, with the associated weight α determined uniquely up to sign.
The GKM theory [12,17] tells us that the T -equivariant cohomology H * T (M ) of a GKM manifold M is determined by the combinatorial data of its 1-skeleton as a subring of the T -equivariant cohomology H * T (M T ) of its T -fixed point set.Indeed, by torus localization, H * T (M ) is embedded into by the pullback homomorphism induced by the inclusion M T ⊂ M .(1) The cohomology groups of Y h vanish in odd degrees.In particular, , where (i, j) denotes the transposition interchanging i and j.
By Theorem 3.2, we thus have the following description of H * T (Y h ), via the restriction map (3.10) res : Recall that the T -equivariant cohomology of the Hessenberg variety X h admits a similar description in [27] (see also [ Comparing (3.11) with (3.12), we find that the ring isomorphism restricts to an isomorphism (3.14) ξ : [6, p.16689] where vp v denotes the action of v on p v by (3.16).Indeed, if p v ≡ p v•(i,j) modulo t i − t j , then Remark 3.5.The T -weights on the tangent spaces of the fixed points are well defined only up to sign.We use the sign choices in Theorem 3.3 (4).
(See [6, (10) and p.16687].) Remark 3.6.One can check that the isomorphism ξ is in fact the natural one given by ) is acted on by T × T by left and right multiplications so that Y h = T \Z h and X h = Z h /T .If we identify S n with the group of permutation matrices in U (n) so that T S n = S n T , one can easily check that (3.13) is given by the natural isomorphism 5. An S n -action on H * (Y h ).In [27], Tymoczko defined an action of S n on the cohomology of the Hessenberg variety X h whose Frobenius characteristic turns out to coincide with the chromatic quasisymmetric function (2.2) by [8,14].Similarly, there is a natural action of S n on the cohomology of Y h .Let us first recall the dot action on H * (X h ).Consider the S n -action on ).This is called the dagger action in [21].
Using this dagger action, we have the following.
where ch is the Frobenius characteristic from the ring of representations of symmetric groups S n onto the ring Λ of symmetric functions [20, §I.7].For a Hessenberg function h : [n] → [n], we define We identify S n with the group of permutation matrices in U (n).In this case, we have a left (resp.right) action of S n on X (resp.Y ) by left (resp.right) multiplication of permutation matrices.
Let V i denote the rank i tautological vector bundle on the flag variety X.As the cohomology ring of X is generated by the line bundles V i /V i−1 and the S n action preserves the line bundles, we find that the action of S n on H * (X) is trivial (cf.[27, Proposition 4.2], [16,Example 2.12]).As the action of S n on the equivariant line bundles V i /V i−1 permutes the equivariant weights, we find that the T -equivariant cohomology of X is where S n acts trivially on H * (X) and by (3.16) on . By Remark 3.6 and (3.13), we find that Applying the Frobenius characteristic to (3.19), we obtain because ch q (H * (X)) = [n] q !. From this, it follows that K n is uniquely determined by the inductive formula (2.4).Indeed, by Lemma 3.11 (2) below.
Example 3.9 immediately implies the following.
The remaining two conditions in Theorem 2.6 will be proved in §3.6 and in §5.
Then {f n } n≥0 satisfies the following: (1) ) spanned by monomials generated by less than (resp.precisely) n variables.Then, we have Moreover, W n admits a decomposition by the number of generating variables where Ind Sn S i ×S n−i W ′ n−i is the induced representation of W ′ n−i as an S i × S n−i representation with a trivial S i -action.Since ch is multiplicative with respect to the multiplication on representations of symmetric groups given by taking the induced representation of tensor products ([20, I, (7.3)]), we have by (3.21).This proves (1).
When n = 0, (2) trivially holds since q 0 f 0 = 1 = e 0 f 0 .Assume that n ≥ 1 and (2) holds for every m < n, so that Then, by the assertion (1), (3.23) and (3.22), we have where the last equality holds by A by-product of Lemma 3.11 is an elementary proof of the following.
Corollary 3.12.[21, Lemma 5.0.1 (2)] Let ω : Λ ∼ = Λ be the involution of the ring Λ which interchanges e n and h n .The we have Proof.Let f ′ n := ωf n for n ≥ 0. We use induction on n.The assertion trivially holds for n = 0. Assume that n ≥ 1 and f m (q −1 ) = (−q) m f ′ m (q) for m < n.By applying ω to Lemma 3.11 (2), we have where the third and the fourth equalities hold by the induction hypothesis and Lemma 3.11 (1) respectively.Therefore the corollary holds for all n.
Corollary 3.12 is a key ingredient in the proof of the parlindromicity of P(h) up to the involution ω by Masuda and Sato in [21] which says that where sgn denotes the sign representation.In fact, this parlindromicity (3.24) follows from Corollary 3.12 and the modular law (Theorem 5.3) as follows: First one can immediately check (3.24) for the isospectral manifold Y in (3.1) using Corollary 3.12 and (3.20).Next the modular law enables us to deduce (3.24) for Y h from the palindromicity for Y .Remark 3.13.Complete homogeneous symmetric functions h n are used only in Lemma 3.11 and the proof of Corollary 3.12.We hope these not to be confused with Hessenberg functions.
3.6.Connectedness and multiplicativity.In this subsection, we give a criterion for connectedness of twin manifolds, and check the multiplicative property of the function P in Definition 3.8.

Lemma 3.14. For a Hessenberg function
Proof.This is given by the explicit diffeomorphism where Y h ′ ,I ∼ = Y h ′ denotes the twin manifold with spectrum {λ i | i ∈ I} and Y h ′′ ,I c ∼ = Y h ′′ is defined similarly.
Proof.Under the identifications (3.5) and (3.9), the isomorphism (3.25) restricts to an isomorphism on T -fixed point sets where w I is an element of S n sending [j] and [j] c to I and I c respectively such that w I | [j] and w I | [j] c are increasing.This induces an isomorphism and p v = 0 otherwise for each I.This is equivariant under the induced actions of S I × S I c and S n from (3.17) for each component.Furthermore, one can easily see that the above isomorphism restricts to an isomorphism Taking quotients by the submodules generated by m, H * (Y h ) is the induced representation of the S j × S n−jrepresentation H * (Y h ′ ) ⊗ H * (Y h ′′ ).Since the Frobenius characteristic ch is multiplicative with respect to tensor products (cf.[20, I, (7.3)]), we have P(h) = P(h ′ )P(h ′′ ) as desired.
By comparing the twin manifolds geometrically, we will prove below that the last condition (3) in Theorem 2.6 is also satisfied for P(h) (cf.Theorem 5.3 below) and hence give a direct proof of the LLT-SW conjecture (cf.Theorem 5.4).

Geometry of twin manifolds
In this section, we compare the twin manifolds associated to a modular triple h = (h − , h, h + ) of Hessenberg functions in Definition 2.3.More precisely, we construct a manifold Y h , called the roof manifold of h, together with maps where pr 2 is a smooth fibration with fiber Y h and π is the blowup along the submanifold Y h − of complex codimension 2. The modular law (4.1) for P(h) then follows immediately from the blowup formula for π and the spectral sequence for pr 2 .
4.1.Roof of a triple.In this subsection, we define the roof be the embedding given by the matrix multiplication.(1) When h(j) = h(j + 1) and h −1 (j) = {j 0 } for some 1 ≤ j 0 < j < n, let 1 ≤ r ≤ n − j be any integer such that In this case, we define ) and to be the quotients of Y h × U (r + 1) and Y h − × U (r + 1) by the free actions of U (1) × U (r) given by (2) When h(j) + 1 = h(j + 1) = j + 1 and h −1 (j) = ∅ for some j < n, let 1 ≤ r ≤ j be any integer such that In this case, similarly we define 1) U (r + 1) and given by the actions (4.2) for B ∈ U (r) × U (1), via We call a triple h = (h − , h, h + ) in ( 1) and ( 2) a triple of type ( 1) and (2) respectively.
Remark 4.2.Note that when r = 1, h is a modular triple in Definition 2.3.Also note that triples of type ( 1) and ( 2) with the same r are dual to each other via the involution map h → h t on the set of Hessenberg functions, where for a Hessenberg function h : One can easily see that h is a triple of type (1) (resp.( 2)) if and only if h t is a triple of type (2) (resp.( 1)).These are our geometric relations among the twin manifolds that will give us the modular law (4.1).
To prove (*) and (**), we have to construct maps among the manifolds the second projection induces the fiber bundle of fiber bundles over P r .Moreover, we have the map Similarly, we have the map These are well defined, since Y h + and Y h − are invariant under the right multiplication of ι(U (r + 1)) by (3.5).
Using the notation of (3.6), let f h : Y h + → C r+1 be a map defined by is the transversal vanishing locus of f h by (3.5) and (3.6).This induces a map (4.9) Example 4.4.Let h be as in Example 4.3 (1).Then, we have The map (4.9) is given by The following propositions illustrate the geometry of Y h for a triple h which is very similar to that of a triple of Hessenberg varieties X h studied in [16, §3.3], via blowups and projective bundles.
Proof.Note that Y h − is invariant under the action of ι(U (r + 1)).Hence the map (T g, [A]) → [T gι(A) −1 , A] is the well defined inverse.
Proposition 4.6.The map induced by (4.4) and (4.6) is an embedding of Y h onto the submanifold defined by In particular, (4.9) fits into the diagram where pr 2 is a smooth fibration with fiber Y h .(4.9) Proof.Since Y h + is invariant under the action of ι(U (r + 1)) for ι defined in Definition 4.1, the same argument in the proof in Proposition 4.5 proves that Y h + × P r is isomorphic to the quotients Y h + × U (1)×U (r) U (r + 1) or Y h + × U (r)×U (1) U (r + 1) defined in the same manner as in Definition 4.1 (1) and ( 2) respectively.Under this isomorphism, (4.11) is induced from the canonical inclusion Y h ⊂ Y h + , in particular it is an embedding.
Under the isomorphisms (4.3) which send the a = [A] to the class represented by the first (resp.last) row vectors of A, this is equivalent to that the vector f h (y) is parallel to a vector v representing a = [v], or equivalently, f h (y) ∈ Cv.The last assertion is immediate.Proposition 4.7.
(1) The left square in the commutative diagram (3) π is the trivial P r -bundle over Y h − via (4.10).( 4) The normal bundle of  is isomorphic to the pullback of the tautological complex line bundle O P r (−1) via (4.10), as a real vector bundle. Proof.
(1) and (3) follow from (4.8), which implies the vanishing of (4.14) for every A ∈ U (r + 1), so that follows from Proposition 4.6.( 4) follows from the fact that E h is the vanishing locus in Y h of the map when h is of type ( 1) and when h is of type (2) respectively, and the fact that E h ⊂ Y h is submanifold of real codimension two.The complex line bundles C × U (1)×U (r) U (r + 1) and C × U (r)×U (1) U (r + 1) are the tautological complex line bundle O P r (−1) over P r by (3.8), via the isomorphisms (4.3).
This induces a natural T -action on P r via (4.3), which coincides with the componentwise multiplication of t ′ .
It is straightforward to see that all the morphisms in Propositions 4.5, 4.6 and 4.7 are T -equivariant.By Proposition 4.6, pr 2 is a fiber bundle with fiber Y h .By Proposition 3.4 and (3.18), the odd degree parts of H * (Y h ) and H * T (Y h ) vanish.Hence the spectral sequence for pr 2 degenerates and we have isomorphisms Letting γ = c 1 (O P r (−1)), we have a ring isomorphism H * (P r ) ∼ = Q[γ]/(γ r+1 ) and the second isomorphism in (4.17) is in fact the inverse of (4.18) by Proposition 4.5 above.Here we are abusing the notation by denoting the pullback of γ to E h by γ to simplify the notation.Similarly as in (4.17), we have By Proposition 4.7, we have the following blowup formula.
Proposition 4.9.Let h = (h − , h, h + ) be as above.Then the map ) is an isomorphism, where e(N  ) denotes the Euler class of the normal bundle N  of the canonical inclusion  : E h ֒→ Y h and  * denotes the Gysin homomorphism induced by .Similarly, the map Proof.We will show that  * : H * ( Y h ) → H * (E h ) induces an isomorphism (4.22) Coker (π * ) using Propositions 4.5 and 4.7.Then a splitting of a short exact sequence Proposition 4.7 (4) and this corresponds to β i via (4.18).
In the rest of the proof, we show that (4.22) is an isomorphism.By (4.17), 1) and ( 2) in Proposition 4.7, we have a commutative diagram of exact sequences (4.23) 0 without two 0's at the top, for each k, where the two rows are parts of the long exact sequences of cohomology with compact supports.By (4.10) and the five lemma, π * − and π * are injective respectively.Then one can check that the horizontal arrow at the bottom is an isomorphism by a diagram chase.In particular, we have in (4.22),where the second isomorphism follows by Proposition 4.5.
By the same argument with the equivariant cohomology instead of the ordinary cohomology, the second assertion also follows, since all maps used in the argument above, including π − , π and , are T -equivariant.4.4.S n -representations on the cohomology of the roof.In this subsection, we prove the following.
) T where all the inclusions are equalities since By (4.17) and (4.19), the odd degree part of H * ( Y h ) vanishes and hence the roof Y h is equivariantly formal.In particular, the restriction map (4.25) res : Since res in (4.25) and (π T ) * are S n -equivariant as π T (v, σ) = v under the isomorphism (4.24), it is enough to show that the T -equivariant Euler Remark 4.12.More precisely, the T -equivariant Euler class of O P r (−1)| σ i above is t j 0 −t j+i for type (1) t j−r+1+i −t h(j+1) for type (2) respectively, up to sign, by the proof of Proposition 4.7 ( 4) and (3.8).
Example 4.13 (r = 1).Let h = (h − , h, h + ) be a modular triple of type (1).In particular, r = 1.Then, the inclusions induce the short exact sequence which is not a zero divisor.This gives the decomposition (4.27) ) the isomorphism (4.27) now reads as ), which preserves the submodule generated by m.One can also immediately check that it is S n -equivariant since e T (N  ) and φ * • ı * are.Hence, we have The arguments used in the above example easily extend to a more general setting in the following lemma.
of (4.19) and (4.25) is S n -equivariant where S n acts trivially on H * (P r ).Furthermore, the above isomorphism preserves the submodule generated by m.
Proof.Suppose h is of type (1).Let / / ∼ = 0 H * T (pr −1 2 (σ k )) for each 1 ≤ k ≤ r, which is split.Indeed,  * k • ( k ) * is equal to the multiplication by α k := e T (N which is not a zero divisor. The short exact sequence (4.28) gives us the isomorphism as graded vector spaces.Furthermore, there is an explicit isomorphism k is the projection map induced by the inclusion S n × {σ k } ֒→ S n × {σ i } i≤k , which is S n -equivariant.On the other hand, the isomorphism (4.29) is T -equivariant with respect to the usual T -action on Y h composed with the interchange of t j and t j+k .This completes the proof for h of type (1).
For h of type (2), consider the coordinate planes H k ∼ = P k in P r spanned by the coordinate points σ r , • • • , σ r−k+1 and the induced T -equivariant filtration Then the proof is parallel to that for h of type (1) and we omit the detail.

Unicellular LLT polynomials and twin manifolds
Let h = (h − , h, h + ) be a triple of Hessenberg functions in Definition 4.1.Let Y h denote the roof manifold which is a Y h -fiber bundle over P r by Proposition 4.6 and also the blowup of Y h + along Y h − by Proposition 4.7.
In this section, we will apply the geometry of the twin manifolds associated to h to compare the cohomology of the twin manifolds.In particular, we will establish the modular law (2.3) for a modular triple h (when r = 1).
Moreover, we have canonical S n -equivariant isomorphisms (5.4) for a modular triple h.

5.2.
Unicellular LLT and twins.The chromatic quasisymmetric functions and the representations of S n on the cohomology of Hessenberg varieties are related by the Shareshian-Wachs conjecture [23], proved in [8,14], (5.5) F(h) := k≥0 ch(H 2k (X h ))q k = ω csf h (q) where the S n -action on H * (X h ) is given by the dot action (3.15) and ω denotes the involution of Λ which interchanges each Schur function with its transpose.

( 4 )
Every 2-dimensional T -invariant closed submanifold which is the union of T -orbits of dimension at most one, has a T -fixed point.By (2)-(4) in Definition 3.1, the 1-skeleton of M , which is by definition the union of 0 or 1-dimensional T -orbits, is the union of M T and T -invariant 2-spheres.The induced T -action on each T -invariant 2-sphere S 2 ∼ = P 1 is of the form e runs over the set of T -invariant 2-spheres in M , {y e , y ′ e } is the set of T -fixed points in e and α e is the T -weight associated to e.

3. 4 .
Equivariant cohomology of Y h .For the T -action on Y h by (3.7), we may use the GKM theory to investigate the cohomology of Y h .Theorem 3.3.Y h is a GKM manifold by the following.
−b   where I k denotes the k × k identity matrix for each k.When a = b, let ι a := ι I .For disjoint I = [a, b] and J = [c, d] ⊂ [n], we denote by

Definition 4 . 1 .
Given a Hessenberg function h : [n] → [n], consider a triple h = (h − , h, h + ) of Hessenberg functions defined by either of the following.

Lemma 4 . 14 .
The composition r i=0 H k ∼ = P k be the coordinate plane in P r spanned by the coordinate points σ0 , • • • , σ k .This induces a T -equivariant filtration Y h ∼ = Y 0 ⊂ • • • ⊂ Y r = Y h of Y h , where Y k are given by Y k := pr −1 2 (H k ) ∼ = {(y, [v]) ∈ Y h + × H k : f h (y) ∈ Cv}.Equivalently, Y k is the (smooth) intersection of Y h and Y h + ×H k in Y h + ×P r .Let  k : Y k−1 ⊂ Y k denotethe inclusion.Associated to this filtration, there is a Gysin sequence (4.28) 0 / / H * −2 T ( Y k−1 )

( 4 .
29)Y h ∼ = Y 0 ∼ = − − → pr −1 2 (σ k ) = {y ∈ Y h + : f h (y)∈ Ce k } which sends T g to T g • (j, j + k) where (j, j + k) denotes the permutation matrix in U (n) associated to the transposition (j, j + k) ∈ S n .Hence, it remains to show that ( k ) * and ρ k are S n -equivariant.To see this, observe that (4.28) fits into the commutative diagram0 / / H * −2 T ( Y k−1 ) H * T (pr −1 2 (σ k ) T ) / / 0of short exact sequences, where the vertical maps are all injective and the bottom row is induced by
[6]X h ), as it preserves the submodule generated by m := (t 1 , • • • , t n ).This is called Tymoczko's dot action.Definition 3.7.[6]By the isomorphism ξ in (3.14), the dot action on H 12), it is straightforward to check that this action preserves the subring H * T (X h ) ⊂ H * T (X T h ).Hence we have an induced action of S n on H * T (X h ) which in turn defines an S n -action onH * (X h ) ∼ = H * T (X h )/mH * * T (X h ) pulls back to an action of S n on H * T (Y h ) defined by (3.17) (µ, (p v ) v∈Sn ) → (p µ −1 v ) v∈Sn for µ ∈ S n .As this preserves mH * T (Y h ), (3.17) defines an action of S n on(3.18) 4.3.Cohomology of the roof.In this subsection, we compare the cohomology and T -equivariant cohomology of the twin manifolds Y h − , Y h and Y h + associated to a triple h = (h − , h, h + ) by Propositions 4.6 and 4.7.First observe that we have natural T -actions on Y h and E h as follows.
Proposition 4.10.There are S n -actions on H * T ( Y h ) and H * ( Y h ) so that (4.17), (4.19), (4.20) and (4.21) are all isomorphisms of S n -representations.Proof.The statement for the ordinary cohomology follows from that for the equivariant cohomology by (3.18).By using the S n -actions on H * T (Y h ), H * T (Y h − ) and H * T (Y h + ), we can define two actions of S n on H * T ( Y h ) by the isomorphisms (4.19) and (4.21).We have to show that the two actions of S n coincide.Note that the set of T -fixed points in Y h is (P r ) T ∼ = S n × {σ i } 0≤i≤r where σ i denote the i-th coordinate points.Indeed, by (4.10) and (4.11), we have