Unicellular LLT polynomials and twin manifolds

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#Unicellular LLT polynomials

The unicellular LLT polynomials are the one-cell-per-component subfamily of the LLT polynomials. In the tuple model, this means that the indexing tuple consists of single boxes. A more convenient description, observed by E. Carlsson and A. Mellit, and independently in joint work with G. Panova, uses unit interval graphs and their area sequences [CM17, AP18].

#The graph-coloring model

Let $\avec=(a_1,\dotsc,a_n)$ be an area sequence of a Dyck path, and let $\Gamma_\avec$ be the corresponding natural unit interval graph on vertex set $[n].$ We use the same graph and ascent statistic as on the chromatic quasisymmetric function page. The vertices are ordered as \[1\lt 2\lt \dotsb \lt n.\] For a coloring $\kappa:[n]\to \setP,$ an edge $\{i,j\}$ with $i\lt j$ is an ascent of $\kappa$ if $\kappa(i)\lt\kappa(j).$ We write $\asc(\kappa)$ for the number of such edges. The unicellular LLT polynomial is \[\LLT_\avec(\xvec;q) \coloneqq \sum_{\kappa:[n]\to\setP} x_{\kappa(1)}\dotsm x_{\kappa(n)}q^{\asc(\kappa)}.\] All colorings are allowed. This is the main difference from the corresponding chromatic quasisymmetric function.

The graph-coloring model is equivalent to the one-cell tuple definition of LLT polynomials in the Haglund–Haiman–Loehr model [HHL05]. The explicit bijection between tuples of single boxes and Dyck diagrams is described in [AP18].

Example (Bijection between tuples and Dyck diagrams).

The $3$-tuple of skew shapes $3211/211,$ $4221/321,$ $421/31$ gives a unicellular LLT polynomial. The boxes are labeled in reading order, and the corresponding boxes are labeled in the Dyck diagram with area sequence $011101012.$

$ 3 $ $ 5 $ $ 9$ $ 1 $ $ 4 $ $ 8 $ $ 2$ $ 6 $ $ 7 $ $ 9$ $ 8$ $ 7$ $ 6$ $ 5$ $ 4$ $ 3$ $ 2$ $ 1$

In general, if the tuple consists of possibly disconnected ribbons of size at most $k,$ then the entries in the corresponding area sequence are less than $k.$

#Relation with chromatic quasisymmetric functions

The same graph and ascent statistic control both families. The chromatic quasisymmetric function imposes properness; the LLT polynomial does not. In [CM17], E. Carlsson and A. Mellit prove the plethystic relation \[(q-1)^{-n}\LLT_\avec[\xvec(q-1);q] = \chrom_{\Gamma_\avec}(\xvec;q).\]

#Basic properties

The complete graphs on $n$ vertices are unit interval graphs. For the complete graph $K_n,$ one has \[\LLT_{K_n}(\xvec;q) = \macdonaldH_{(n)}(\xvec;q,t) = \sum_{T\in\SYT(n)}q^{\cocharge(T)}\schurS_{\lambda(T)}(\xvec).\] Here $\macdonaldH_{(n)}(\xvec;q,t)$ is a modified Macdonald polynomial, and the last sum is a modified Hall–Littlewood polynomial.

In joint work with G. Panova, we prove that for any area sequence $\avec$ of length $n,$ \[\omega \LLT_{\avec}(\xvec;q) = q^{a_1+\dotsb+a_n}\LLT_{\avec^T}(\xvec;1/q),\] where $\avec^T$ denotes the transpose of the Dyck diagram. From the definition, \[\LLT_{\avec^T}(x_1,x_2,\dotsc,x_n;q) = \LLT_{\avec}(x_n,x_{n-1},\dotsc,x_1;q),\] and since the polynomials are symmetric, it follows that $\LLT_{\avec^T}(\xvec;q)=\LLT_{\avec}(\xvec;q).$

The unicellular LLT polynomials are also evaluations of certain traces of Hecke algebras; see [MSW24]. These traces are closely connected with Kazhdan–Lusztig polynomials.

#Recursions

Several recursions and linear relations are known for unicellular LLT polynomials. For example, S. J. Lee gives a three-term recursion in [Lee20]. Short bijective proofs of these recursions are given in [Ale20].

Lee’s recursions are generalized further by C. Miller; see [Eq. (2.2), Mil19]. For more generalizations and results on linear relations between LLT polynomials, see [Tom20] and [Kea21].

#Schur expansion

It is a major open problem to find a combinatorial rule for the Schur expansion of $\LLT_\avec(\xvec;q).$ For so-called melting lollipop graphs, J. Huh, S. J. Nam, and M. Yoo give a combinatorial formula for the Schur expansion [HNY20]. V. Wang extends this direction to two-headed melting lollipop graphs by giving a formula expanding the corresponding unicellular LLT polynomials into ribbon Schur functions [Wan25].

Example (Schur expansions for length 3 area sequences).

For area sequences of length $3,$ one has \[\begin{aligned} \LLT_{000}(\xvec;q)&=\schurS_{3}+2\schurS_{21}+\schurS_{111}, \\ \LLT_{001}(\xvec;q)&=\schurS_{3}+(1+q)\schurS_{21}+q\schurS_{111}, \\ \LLT_{011}(\xvec;q)&=\schurS_{3}+2q\schurS_{21}+q^2\schurS_{111}, \\ \LLT_{012}(\xvec;q)&=\schurS_{3}+(q+q^2)\schurS_{21}+q^3\schurS_{111}. \end{aligned}\]

Since LLT polynomials are $q$-deformations of Littlewood–Richardson coefficients [LT00], the unicellular case suggests the following standard-tableau formulation.

Conjecture (See [LT00]).

There is a statistic $c_\avec(T)$ on standard Young tableaux, depending on $\avec,$ such that \[\LLT_{\avec}(\xvec;q) = \sum_{T\in\SYT(n)}q^{c_\avec(T)}\schurS_{\lambda(T)}.\] By comparing the coefficients of $x_1x_2\dotsm x_n$ on both sides, one obtains \[\sum_{\sigma\in\symS_n} q^{\asc_\avec(\sigma)} = \sum_{\lambda\vdash n}f^\lambda \sum_{T\in\SYT(\lambda)}q^{c_\avec(T)},\] where the left-hand side sums over all vertex colorings of $\Gamma_\avec$ using the $n$ distinct colors $1,\dotsc,n,$ and $f^\lambda$ is the number of standard Young tableaux of shape $\lambda.$

Theorem (See J. Blasiak [Bla16]).

Whenever $\nuvec=(\nu^1,\nu^2,\nu^3)$ is a $3$-tuple of possibly disconnected skew shapes, $\LLT_{\nuvec}(\xvec;q)$ is Schur-positive with an explicit combinatorial formula. In the unicellular case, this includes the area sequences indexing $\LLT_{\avec}(\xvec;q)$ with $\max_i a_i\leq 2.$

In his PhD thesis, C. Miller shows that LLT polynomials with bandwidth at most $3$ are $3$-Schur-positive, which is stronger than Schur positivity [Mil19]. The bandwidth-at-most-two case was treated earlier by S.J. Lee [Lee20].

S. Cho and J. Huh give a combinatorial formula for the hook coefficients, that is, the coefficients of $\schurS_{\lambda}$ where $\lambda$ has the form $(m,1,1,\dotsc,1).$

#Power-sum expansion

From the power-sum expansion of chromatic quasisymmetric functions, one can derive a formula for $\LLT_\avec(\xvec;q)$ in the power-sum basis. Alternatively, the general machinery in [AS19] gives the following shifted power-sum expansion; see also the related result on P-partitions.

Given a poset $P$ on $n$ elements, let $\opsurj(P)$ be the set of order-preserving surjections $f:P\to[k]$ for some $k.$ The type of such a surjection is \[\alpha(f)\coloneqq (|f^{-1}(1)|,|f^{-1}(2)|,\dotsc,|f^{-1}(k)|).\] Let $\opsurj_\alpha(P)\subseteq\opsurj(P)$ be the set of surjections of type $\alpha,$ and let $\opsurj^\ast_\alpha(P)\subseteq\opsurj_\alpha(P)$ be the set of surjections $f\in\opsurj_\alpha(P)$ such that, for each $j\in[k],$ $f^{-1}(j)$ is an induced subposet of $P$ with a unique minimal element.

Theorem (Alexandersson–Sulzgruber [AS19]).

Let $O(\avec)$ denote the set of orientations of the unit interval graph given by $\avec.$ Then \[\omega \LLT_{\avec}(\xvec;q+1) = \sum_{\theta\in O(\avec)}q^{\asc(\theta)} \sum_{\lambda\vdash n} \frac{\powerSum_\lambda(\xvec)}{z_\lambda} |\opsurj^\ast_\lambda(P(\theta))|,\] where $P(\theta)$ is the poset on $[n]$ given by the transitive closure of the ascending edges in $\theta.$

Example (Power-sum expansion for $\avec=0121$ with orientations).

The area sequence $\avec=(0,1,2,1)$ gives the expansion \[\begin{aligned} \omega \LLT_{0121}(\xvec;q+1) = {}&(2q^3+q^4)\frac{\powerSum_4}{z_4} +(4q^2+4q^3+q^4)\frac{\powerSum_{31}}{z_{31}} \\ &+(2q^2+2q^3+q^4)\frac{\powerSum_{22}}{z_{22}} +(8q+12q^2+6q^3+q^4)\frac{\powerSum_{211}}{z_{211}} \\ &+(24+48q+34q^2+10q^3+q^4) \frac{\powerSum_{1111}}{z_{1111}}. \end{aligned}\]

The following two orientations $\theta_1,\theta_2$ contribute to $2q^3\powerSum_{22}/z_{22}.$ By convention, only the ascending edges are shown.

$ \to $ $ 4$ $ \to $ $ 3$ $ \to $ $ 2$ $ 1$ $ \to $ $ 4$ $ \to $ $ 3$ $ \to $ $ 2$ $ 1$

For each orientation, the order-preserving surjection $f$ defined by $f(1)=f(2)=1$ and $f(3)=f(4)=2$ is the only one satisfying the conditions for membership in $\opsurj^\ast_{22}(P(\theta)).$ The minimal elements of $f^{-1}(1)$ and $f^{-1}(2)$ are $1$ and $3,$ respectively.

For $\theta_2,$ the map $g(1)=g(3)=1$ and $g(2)=g(4)=2$ is also an order-preserving surjection of type $(2,2),$ but the induced subposet $g^{-1}(2)\subset P(\theta_2)$ has both elements as minimal elements. Hence $g$ is not an element of $\opsurj^\ast_{22}(P(\theta_2)).$

A similar-looking formula for chromatic symmetric functions is due to O. Bernardi and P. Nadeau [Prop. 5.2, BN20].

#Elementary symmetric expansion

Together with G. Panova, we conjectured in [AP18] that for any unit interval graph $\avec,$ the polynomial $\LLT_\avec(\xvec;q+1)$ is positive in the elementary symmetric function basis. The formula below was later proved by P. Alexandersson and R. Sulzgruber.

Let $\theta\in O(\avec),$ and for each $i\in[n]$ define the highest reachable vertex by \[\mathrm{hrv}(i)\coloneqq \max\{j\in[n]: i\leq_{P(\theta)}j\}.\] Here $P(\theta)$ is the poset generated by the ascending edges of $\theta.$ Note that $\mathrm{hrv}(i)\geq i$ for all $i\in[n].$ Finally let \[\pi(\theta)=(|\mathrm{hrv}^{-1}(1)|,\dotsc,|\mathrm{hrv}^{-1}(n)|),\] with zero parts omitted when it is used as an index of $\elementaryE.$

Theorem (Alexandersson–Sulzgruber [AS20]).

For any unit interval graph $\avec$ with $n$ vertices, \[\LLT_\avec(\xvec;q+1) = \sum_{\theta\in O(\avec)}q^{\asc(\theta)} \elementaryE_{\pi(\theta)}(\xvec).\]

This is an analogue of the Stanley–Stembridge conjecture for chromatic quasisymmetric functions. A similar formula holds for vertical-strip LLT polynomials. The path graph case appears in [AP18], and the complete graph case follows from a recursion. For melting lollipop graphs, [Ale20] gives a similar-looking $\elementaryE$-expansion.

Example

In the diagram with $\avec=012223,$ the orientation $\theta\in O(\avec)$ is shown by marking the ascending edges with $\to.$

$ \to $ $ 6$ $ 5$ $ \to $ $ 4$ $ \to $ $ 3$ $ \to $ $ 2$ $ 1$

The highest reachable vertices for $1,2,\dotsc,6$ are $4,2,4,4,6,6,$ respectively. Hence $\pi(\theta)=(0,0,1,0,3,0,2),$ and this orientation contributes $q^5\elementaryE_{321}(\xvec)$ to $\LLT_{012223}(\xvec;q+1).$

The full expansion is \[\begin{aligned} \LLT_{012223}(\xvec;q+1) = {}&(24q^5+60q^6+62q^7+33q^8+9q^9+q^{10})\elementaryE_6 \\ &+(8q^4+12q^5+6q^6+q^7)\elementaryE_{33} \\ &+(20q^4+36q^5+25q^6+8q^7+q^8)\elementaryE_{42} \\ &+(40q^4+96q^5+94q^6+46q^7+11q^8+q^9)\elementaryE_{51} \\ &+(4q^3+4q^4+q^5)\elementaryE_{222} \\ &+(36q^3+50q^4+24q^5+4q^6)\elementaryE_{321} \\ &+(34q^3+69q^4+56q^5+21q^6+3q^7)\elementaryE_{411} \\ &+(19q^2+17q^3+4q^4)\elementaryE_{2211} \\ &+(20q^2+28q^3+15q^4+3q^5)\elementaryE_{3111} \\ &+(10q+6q^2+q^3)\elementaryE_{21111} +\elementaryE_{111111}. \end{aligned}\]

A refinement to LLT cumulants imposes certain connectivity requirements [Kow20]; see also [DK22]. Another refinement is given by A. Abreu and A. Nigro in [AN21], where they consider symmetric functions defined as sums over increasing forests. The unicellular LLT polynomials can be expressed as sums over these forests.

#Representation theory

The polynomial $\LLT_\avec(\xvec;q)$ is the Frobenius characteristic of twin cohomology with the dagger action.

#Frobenius characteristic

The polynomial $\LLT_\avec(\xvec;q)$ is symmetric and Schur-positive. Hence it can be interpreted as a graded Frobenius characteristic: \[\LLT_\avec(\xvec;q) = \sum_{d\geq 0}q^d\frobChar(M_\avec^d)\] for some graded $\symS_n$-module \[M_\avec=\bigoplus_{d\geq 0}M_\avec^d.\] The question is to construct a natural module $M_\avec$ with this graded Frobenius characteristic.

M. Guay-Paquet used Hopf-algebra methods to connect the Shareshian–Wachs theorem with unicellular LLT polynomials [Gua16]. The twin-manifold story gives a cohomological realization: \[\sum_{d\geq 0} q^d \frobChar\left(H^{2d}(\mathrm{Twin}_\avec)\right) = \LLT_\avec(\xvec;q).\] This is the LLT analogue of the Hessenberg dot-action theorem. Y.-H. Kiem and D. Lee give a direct geometric proof of this statement by comparing twin manifolds through a roof manifold and showing that their cohomology satisfies the modular law for unicellular LLT polynomials [Thm. 5.4, KL24]. This proof does not pass through the Shareshian–Wachs conjecture for chromatic quasisymmetric functions. The Hilbert series of this graded module is obtained by pairing with $\completeH_{1^n}:$ \[\left\langle \LLT_\avec(\xvec;q),\completeH_{1^n}\right\rangle = \sum_{\sigma\in\symS_n}q^{\asc(\sigma)}.\] The right-hand side is the ascent enumerator over colorings using all $n$ colors exactly once, equivalently over permutations.

A complementary finite-group interpretation is due to L. Gagnon. He realizes vertical-strip LLT polynomials by inducing characters from the unipotent upper triangular group to $\GL_n(\mathbb{F}_q)$ and then recording the unipotent constituents [Thm. 5.1, Gag24].

#The twin GKM model

The twin-manifold GKM model is close to the regular semisimple Hessenberg GKM model, with two differences.

First, the congruence labels are fixed roots. The fixed points are indexed by permutations $v\in\symS_n,$ and for each edge $\{i,j\}$ of $\Gamma_\avec$ we have a moment-graph edge \[v\longleftrightarrow v s_{ij}.\] However, the GKM divisibility relation is \[p_v-p_{v s_{ij}}\in (t_i-t_j),\] not \[p_v-p_{v s_{ij}}\in (t_{v(i)}-t_{v(j)}).\] Thus the edge label depends on the graph edge $\{i,j\}$ itself, rather than on the image of the edge under the fixed point $v.$

Second, the group action is the dagger action. It acts by moving the fixed-point labels, without permuting the polynomial variables: \[(\sigma\dagger p)_v \coloneqq p_{\sigma^{-1}v}.\] This should be contrasted with Tymoczko’s dot action, where $\sigma$ also acts on the variables $t_1,\dotsc,t_n.$

The equivariant twin GKM module is therefore \[H_T^\ast(\mathrm{Twin}_\avec) = \left\{ (p_v)_{v\in\symS_n} : p_v-p_{v s_{ij}}\in (t_i-t_j) \text{ for all } v \text{ and } \{i,j\}\in E(\Gamma_\avec) \right\}.\] Ordinary cohomology is obtained by setting $t_1,\dotsc,t_n$ to zero: \[H^\ast(\mathrm{Twin}_\avec) \cong H_T^\ast(\mathrm{Twin}_\avec) \otimes_{\setC[t_1,\dotsc,t_n]}\setC.\] The dagger action descends to ordinary cohomology. Taking the graded Frobenius characteristic then gives the unicellular LLT polynomial.

#Non-commutative unicellular LLT polynomials

J.-C. Novelli and J.-Y. Thibon introduce a non-commutative lift of the unicellular LLT polynomials in [Thm. 6.2, NT19]. They show that these are positive in a non-commutative analogue of the Gessel quasisymmetric basis, and that they satisfy a non-commutative plethystic relation with chromatic quasisymmetric functions.

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[LT00]Bernard Leclerc and Jean-Yves Thibon. Littlewood–Richardson coefficients and Kazhdan–Lusztig polynomials. Combinatorial methods in representation theory, 28:155–220, 2000.

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@inproceedings{LeclercThibon2000,
  doi = {10.2969/aspm/02810155},
  url2 = {https://doi.org/10.2969/aspm/02810155},
  publisher = {Mathematical Society of Japan},
  author = {Bernard Leclerc and Jean-Yves Thibon},
  title = {{L}ittlewood--{R}ichardson Coefficients and {K}azhdan--{L}usztig Polynomials},
  booktitle = {Combinatorial Methods in Representation Theory},
  year  = {2000},
  volume = {28},
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  journal = {Adv. Stud. Pure Math.}
}

[Lee20]Seung Jin Lee. Linear relations on LLT polynomials and their k-Schur positivity for $k=2$. Journal of Algebraic Combinatorics, 53(4):973–990, June 2020.

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@article{Lee2020,
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  month = jun,
  publisher = {Springer Science and Business Media {LLC}},
  volume = {53},
  number = {4},
  pages = {973--990},
  author = {Seung Jin Lee},
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}

[Mil19]Christopher R. Miller. On the $k$-Schur positivity of $k$-bandwidth LLT polynomials. UC Berkeley, 2019.

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@phdthesis{Miller2019,
  author       = {Christopher R. Miller},
  school       = {UC Berkeley},
  title        = {On the $k$-{S}chur Positivity of $k$-Bandwidth {LLT} Polynomials},
  url          = {https://escholarship.org/uc/item/26x8z6dz},
  year         = {2019}
}

[MSW24]Alejandro H. Morales, Mark A. Skandera and Jiayuan Wang. LLT polynomials and Hecke algebra traces. Electronic Proceedings in Theoretical Computer Science, 403:150–155, June 2024.

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@article{MoralesSkanderaWang2024,
  title = {{LLT} Polynomials and {H}ecke Algebra Traces},
  volume = {403},
  ISSN = {2075-2180},
  url = {http://dx.doi.org/10.4204/EPTCS.403.31},
  DOI = {10.4204/eptcs.403.31},
  journal = {Electronic Proceedings in Theoretical Computer Science},
  publisher = {Open Publishing Association},
  author = {Morales,  Alejandro H. and Skandera,  Mark A. and Wang,  Jiayuan},
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[NT19]Jean-Christophe Novelli and Jean-Yves Thibon. Noncommutative unicellular LLT polynomials. arXiv:1907.00077, 2019.

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@article{NovelliThibon2019,
Author = {Jean-Christophe Novelli and Jean-Yves Thibon},
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}

[Tom20]Foster Tom. A combinatorial Schur expansion of triangle-free horizontal-strip LLT polynomials. arXiv:2011.13671, 2020.

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@article{Tom2020x,
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  url = {https://arxiv.org/abs/2011.13671},
journal = {arXiv e-prints}
}

[Wan25]Victor Wang. Expanding the unicellular LLT polynomials of two-headed melting lollipops into ribbon Schurs. arXiv:2504.08187, 2025.

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@article{WangLollipopLLT2025x,
  author = {Victor Wang},
  title = {Expanding the unicellular {{L}{L}{T}} polynomials of two-headed melting lollipops into ribbon {S}churs},
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  eprint = {2504.08187},
  url = {https://arxiv.org/abs/2504.08187},
  journal = {arXiv e-prints}
}