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#LLT polynomials

LLT polynomials were originally introduced by A. Lascoux, B. Leclerc, and J.Y. Thibon in [LLT97]. They form a large family of symmetric functions, and can be seen as a \(q\)-deformation of products of skew Schur functions. This family contains the usual Schur polynomials, the skew Schur functions and the family of transformed Hall–Littlewood polynomials. Here this is recorded as a containment relation, since the Schur and skew Schur functions appear as special LLT data rather than as a uniform one-parameter specialization. The original motivation for LLT polynomials was to study certain plethysm coefficients and their relationship with the Fock space representation of the quantum affine algebra \(U_q(\widehat{\mathfrak{sl}_n}).\) There are generalizations to other types, for example by C. Lecouvey [Lec09], and by I. Grojnowski and M. Haiman [GH06]. P. Alexandersson and J. Uhlin introduce skew specialized nonsymmetric Macdonald polynomials and prove that they are symmetric and Schur-positive [AU20]. In the case where the skew shape has no column with more than two boxes, these polynomials specialize to vertical-strip LLT polynomials, giving a combinatorial Schur expansion for this family of LLT polynomials. For an accessible entry point to the affine-Hecke and ribbon-tableau side of the subject, see T. Lam’s work on ribbon tableaux and affine Stanley symmetric functions [Lam05, Lam06], as well as his AIM notes [Lam05]. M. Haiman’s CRM slides on general LLT polynomials [Hai07] give an expository overview of the Grojnowski–Haiman approach to general LLT positivity.

The original definition expresses the LLT polynomials, called spin LLT polynomials, as a sum over so-called ribbon tableaux, weighted with a spin statistic. The authors originally named them ribbon Schur functions, but they are now referred to as LLT polynomials. E. K. Oğuz gives a type-\(B\) analogue in terms of shifted \(k\)-ribbon tableaux [Og20]. The paper constructs a bijection from shifted \(k\)-ribbon fillings to ordinary fillings of a \(\lfloor k/2\rfloor\)-tuple of shapes, called the \(k\)-quotient, and proves that the associated generating functions are symmetric, Schur-positive, and Schur \(Q\)-positive. A supersymmetric version of LLT polynomials, akin to supersymmetric Schur functions, has also been considered; see, for example, [CFYZ+21] for a lattice model generating such supersymmetric LLT polynomials. A. Gitlin and D. Keating give a Yang–Baxter integrable vertex model for supersymmetric LLT polynomials [GK23]. Their partition functions are essentially T. Lam’s super ribbon functions and yield Cauchy-type identities for these polynomials. D. Keating proves equivalences of LLT polynomials through lattice-path partition functions [Kea25]. The result gives a path-model explanation for when different LLT data produce the same symmetric function.

In [LT00], it is proved that the coefficients in the Schur basis of spin LLT polynomials are parabolic Kazhdan–Lusztig polynomials. For the connection with Kazhdan–Lusztig polynomials and the more representation-theoretic viewpoint, see [LR19].

LLT polynomials have a prominent role in the study of diagonal harmonics. See [BHMP+21] for related results in this area.

#Flagged LLT polynomials

J. Blasiak, M. Haiman, J. Morse, A. Pun, and G. H. Seelinger [BHMP+25] introduce flagged LLT polynomials \(G_{\nu,\sigma}.\) These generalize ordinary LLT polynomials in the same way that flagged Schur functions generalize Schur functions, and they have both algebraic and tableau-sum definitions.

One striking feature is that flagged LLT polynomials interact well with a nonsymmetric plethysm operator. This gives positive expansions of modified \(r\)-nonsymmetric Macdonald polynomials into flagged LLT polynomials. The same paper conjectures that flagged LLT polynomials themselves expand positively in Demazure atoms; this would strengthen the usual Schur positivity story for symmetric LLT polynomials.

#Definition (ribbons)

The original definition is given in [LT00]. A modern reference is [Iij13].

Let \(\SSYT^{(k)}(\lambda/\mu)\) be the set of \(k\)-ribbon tableaux of shape \(\lambda/\mu.\) Then the spin LLT polynomial is defined as \[\LLTG^{(k)}_{\lambda/\mu}(\xvec;q) \coloneqq \sum_{T \in \SSYT^{(k)}(\lambda/\mu)} \xvec_T q^{spin(T)}.\]

#Definition (tuples)

M. Bylund and M. Haiman discovered an alternative way to model LLT polynomials. These are indexed by \(k\)-tuples \(\nuvec\) of skew Young diagrams. When \(k=1,\) we recover a single skew Schur function. The Bylund–Haiman model is described in [HHL05], where it is proved that the two models are equivalent. The proof uses a version of the Littlewood map, which sends ribbon tableaux to tuples of semistandard Young tableaux.

Given a \(k\)-tuple of skew shapes \(\nuvec,\) let \[\SSYT(\nuvec) = \SSYT(\nuvec^1) \times \SSYT(\nuvec^2)\times \dotsm \times \SSYT(\nuvec^k)\] where \(\SSYT(\lambda)\) is the set of skew semistandard Young tableaux of shape \(\lambda.\) For \(T = (T^1,T^2,\dotsc,T^k) \in \SSYT(\nuvec),\) let \(\xvec^T\) denote the product \(\xvec_{T^1}\dotsm \xvec_{T^k}\) where \(\xvec_{T^i}\) is the usual weight of the semistandard Young tableau \(T^i.\) Given a cell \(u = (r,c)\) in a skew diagram, with row \(r\) and column \(c,\) the content of \(u\) is defined as \(c(u)\coloneqq c-r.\)

Entries \(T^i(u) \gt T^j(v)\) in a tuple form an LLT inversion if either

  • \(i \lt j\) and \(c(u) = c(v),\) or

  • \(i \gt j\) and \(c(u) = c(v)-1.\)

Finally, we can define the LLT polynomial \[\LLT_\nuvec(\xvec;q) \coloneqq \sum_{T \in \SSYT(\nuvec)} q^{\inv(T)} \xvec_T\] where \(\inv(T)\) is the total number of inversions appearing in \(T.\) When all shapes in \(\nuvec\) consist of single boxes, we get the family of unicellular LLT polynomials. Note that \(\LLT_\nuvec(\xvec;1) = \prod_{i} \schurS_{\nuvec^i}(\xvec),\) a product of skew Schur polynomials.

Example

Tuples \(\nuvec\) are traditionally illustrated in the French notation when dealing with LLT polynomials. Below, the \(3\)-tuple \(32/1,\) \(31,\) \(33/21\) of skew shapes is illustrated. Note that boxes with same content are placed on the same diagonal (oriented up-right). The numbers indicate the reading order, which controls the LLT inversion statistic.

If \(F\) is a filling of the diagram, we have for example inversions if \(F(1)\lt F(2)\) or \(F(2)\lt F(3).\)

            $\square $ $5 $ $8$             $\square $ $\square $ $11$                         $2 $                 $4 $ $7 $ $10$                         $ 1 $ $3 $               $ \square $ $6 $ $9 $            

The skew boxes have been marked with \(\square.\)

Example (Three one-box shapes).

Let \(\nuvec\) be the tuple consisting of three single boxes, all of content \(0.\) A filling is a word \(abc\) of positive integers, and the LLT inversion statistic is the ordinary inversion number \[\inv(abc)=\#\{(i,j):i\lt{}j,\ a_i\gt{}a_j\}.\] Equivalently, the attacking graph is the complete graph \(K_3\) on the three boxes. Thus \[\LLT_\nuvec(\xvec;q) = \sum_{a,b,c\geq 1} q^{\inv(abc)}x_ax_bx_c.\] Grouping the words by their color multiplicities gives the monomial expansion \[\LLT_\nuvec(\xvec;q) = \monomial_3 +(1+q+q^2)\monomial_{21} +(1+2q+2q^2+q^3)\monomial_{111}.\] In the Schur basis, this is \[\LLT_\nuvec(\xvec;q) = \schurS_3+(q+q^2)\schurS_{21}+q^3\schurS_{111}.\]

The original spin LLT polynomials and the modern LLT polynomials are related as follows. If \(\nuvec\) is the \(k\)-tuple obtained as the \(k\)-quotient of \(\lambda/\mu,\) then there is some \(e,\) depending only on \(\lambda/\mu,\) such that \[q^e \LLT_\nuvec(\xvec;q^{-2}) = \LLTG^{(k)}_{\lambda/\mu}(\xvec;q).\] See [GH06, Prop. 6.17].

#Vertex-model

It has been shown that LLT polynomials can be realized as the partition functions of certain colored fermionic vertex models; see [CGKM21] and [ABW23]. For the connection with fermionic vertex models, see [BBBG20]. The same vertex-model framework also leads to colored corner processes from asymptotics of LLT polynomials [ABW24].

#Properties

#Murnaghan–Nakayama rule

#Pieri rule

#Cauchy identity

A Cauchy identity for the spin LLT polynomials was first given in [Lam05], and a combinatorial proof is given in [Lee05].

In [CGKM21], the authors describe a Yang–Baxter integrable vertex model for LLT polynomials and prove a Cauchy identity. Here, the sum is taken over \(k\)-tuples of integer partitions, \[\sum_{\nuvec} \LLT_\nuvec(x_1,\dotsc,x_n;q) \LLT_{\nuvec^{\mathrm{rot}}}(y_1,\dotsc,y_n;q) = \prod_{i=1}^n \prod_{j=1}^n \prod_{m=0}^{k-1} \frac{1}{1-x_iy_j t^m}\] where \(\nuvec^{\mathrm{rot}}\) rotates each partition 180 degrees and reverses the order of the tuple.

#Fundamental quasisymmetric expansion

It is fairly straightforward from the tuples definition to recover the expansion in the fundamental quasisymmetric basis. Haglund, Haiman, and Loehr show in [HHL05] that \[\LLT_\nuvec(\xvec;q) = \sum_{T \in \SYT(\nuvec)} q^{\inv(T)} \gessel_{D(T)}(\xvec).\] The set \(\SYT(\nuvec)\) consists of all fillings of \(\nuvec\) using entries in \(\setN\) so that each shape has increasing rows and columns. The descent set \(D(T)\) is computed with respect to the reading word. That is, \(i\) is a descent if \(i+1\) appears before \(i\) when reading the boxes of \(\nuvec\) in the reading order.

#Schur positivity

It is proved in [LT00] that \(\LLT_\nuvec(\xvec;q)\) is Schur-positive whenever \(\nuvec\) is a \(k\)-tuple of straight shapes. The proof uses the Fock space representation of \(U_q(\widehat{\mathfrak{sl}_n})\); see [Iij13] for some background.

I. Grojnowski and M. Haiman prove in an unpublished preprint [GH06] that all LLT polynomials \(\LLT_\nuvec(\xvec;q),\) for arbitrary tuples of skew shapes, are Schur-positive. They use Hecke algebras and deep representation theory; their proof does not give a combinatorial formula for the Schur coefficients. For a geometric approach to characters of Hecke algebras, including LLT-related characters and Schur-positivity questions, see [AN25].

At \(t=1,\) single-row Macdonald cumulants are positive sums of vertical-strip LLT polynomials indexed by plane rooted trees [HT26, Thm. 1.3 and Cor. 1.4].

Some partial progress on a combinatorial proof of Schur positivity is given in [Rob13].

The case when the tuple consists of \(k\leq 3\) ribbons is considered by J. Blasiak [Bla16], and an explicit Schur expansion is given. Note that the ribbons need not be connected. This result is used to show that Macdonald polynomials indexed with shapes that have at most three columns are Schur-positive.

In [Tom20], F. Tom gives a combinatorial formula for the Schur expansion of certain vertical-strip LLT polynomials, whenever the underlying unit interval graph is triangle-free; see the page on unicellular LLT polynomials. F. Tom uses a generalization of cocharge. This generalizes an earlier result by P. Alexandersson and J. Uhlin [AU20]. F. Tom also proves that a horizontal-strip LLT polynomial is determined by its weighted graph [Tom23].

#Unicellular LLT polynomials

When all shapes in the tuple consist of single boxes, one obtains the unicellular LLT polynomials. They have a graph coloring model using unit interval graphs, a shifted \(\elementaryE\)-positive orientation formula, and a twin-manifold representation-theoretic interpretation.

#Supersymmetric LLT polynomials

There is a notion of supersymmetric LLT polynomials introduced by T. Lam. See also [BBBG20], where metaplectic symmetric functions are introduced.

#Factorial LLT polynomials

In [ABW23], the authors introduce the notion of factorial LLT polynomials.

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