Shuffle Conjecture Research Articles

Positroid Catalan numbers

Given a (bounded affine) permutation f f , we study the positroid Catalan number C f C_f defined to be the torus-equivariant Euler characteristic of the associated open positroid variety. We introduce a class of repetition-free permutations and show that the corresponding positroid Catalan numbers count Dyck paths avoiding a convex subset of the rectangle. We show that any convex subset appears in this way. Conjecturally, the associated q , t q,t -polynomials coincide with the generalized q , t q,t -Catalan numbers that recently appeared in relation to the shuffle conjecture, flag Hilbert schemes, and Khovanov–Rozansky homology of Coxeter links.

Toric braids and (m,n)-parking functions

We find a geometric interpretation of the Aq,t algebra, the algebra which appeared in the previous work of Erik Carlsson and the author on the proof of the shuffle conjecture. This allows us to construct a representation of “the positive part” of the group of toric braids. Then certain sums over (m,n)-parking functions are related to evaluations of this representation on some special braids. The compositional (km,kn)-shuffle conjecture of Bergeron, Garsia, Leven, and Xin is then shown to be a corollary of this relation.

Schedules and the Delta Conjecture

In a recent preprint, Carlsson and Oblomkov (Affine Schubert calculus and double coinvariants. arXiv preprint 1801.09033, 2018) obtain a long sought-after monomial basis for the ring $$\mathrm{D}\!\mathrm{R}_n$$ of diagonal coinvariants. Their basis is closely related to the “schedules” formula for the Hilbert series of $$\mathrm{D}\!\mathrm{R}_n$$ which was conjectured by the first author and Loehr (Discete Math 298(1–3):189–204, 2005) and first proved by Carlsson and Mellit (A proof of the shuffle conjecture. J Amer Math Soc 31(3):661–697, 2018), as a consequence of their proof of the famous Shuffle Conjecture. In this article, we obtain a schedules formula for the combinatorial side of the Delta Conjecture, a conjecture introduced by the Haglund et al. (Trans Am Math Soc 370(6):4029–4057, 2018), which contains the Shuffle Theorem as a special case. Motivated by the Carlsson–Oblomkov basis for $$\mathrm{D}\!\mathrm{R}_n$$ and our Delta schedules formula, we introduce a (conjectural) basis for the super-diagonal coinvariant ring $$\mathrm{S}\!\mathrm{D}\!\mathrm{R}_n$$ , an $$S_n$$ -module generalizing $$\mathrm{D}\!\mathrm{R}_n$$ introduced recently by Zabrocki (a module for the Delta conjecture. arXiv preprint 1902.08966, 2019), which conjecturally corresponds to the Delta Conjecture.

Schur function expansions and the Rational Shuffle Theorem

Gorsky and Negut introduced operators Qm,n on symmetric functions and conjectured that, in the case where m and n are relatively prime, the expression Qm,n(1) is given by the Hikita polynomial Hm,n[X;q,t]. Later, Bergeron-Garsia-Leven-Xin extended and refined the conjectures of Qm,n(1) for arbitrary m and n which we call the Extended Rational Shuffle Conjecture. In the special case Qn+1,n(1), the Rational Shuffle Conjecture becomes the Shuffle Conjecture of Haglund-Haiman-Loehr-Remmel-Ulyanov, which was proved in 2015 by Carlsson and Mellit as the Shuffle Theorem. The Extended Rational Shuffle Conjecture was later proved by Mellit as the Extended Rational Shuffle Theorem. The main goal of this paper is to study the combinatorics of the coefficients that arise in the Schur function expansion of Qm,n(1) in certain special cases. Leven gave a combinatorial proof of the Schur function expansion of Q2,2n+1(1) and Q2n+1,2(1). In this paper, we explore several symmetries in the combinatorics of the coefficients that arise in the Schur function expansion of Qm,n(1). Especially, we study the hook-shaped Schur function coefficients, and the Schur function expansion of Qm,n(1) in the case where m or n equals 3.

The Delta Conjecture

We conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.

The New dinv is Not So New

In [Duane, Garsia, Zabrocki 2013] the authors introduced a new dinv statistic, denoted ndinv, on the two part case of the shuffle conjecture (Haglund et al. 2005) in order to prove a compositional refinement. Though in [Hicks, Kim 2013] a non-recursive (but algorithmic) definition of ndinv has been given, this statistic still looks a bit unnatural. In this paper we "unveil the mystery" around the ndinv, by showing bijectively that the ndinv actually matches the usual dinv statistic in a special case of the generalized Delta conjecture in [Haglund, Remmel, Wilson 2018]. Moreover, we give also a non-compositional proof of the "ehh" case of the shuffle conjecture (after [Garsia, Xin, Zabrocki 2014]) by bijectively proving a relation with the two part case of the Delta conjecture.

The shuffle conjecture

Walks in the plane taking unit-length steps north and east from ( 0 , 0 ) (0,0) to ( n , n ) (n,n) never dropping below y = x y=x and parking cars subject to preferences are two intriguing ingredients in a formula conjectured in 2005, now famously known as the shuffle conjecture. Here we describe the combinatorial tools needed to state the conjecture. We also give key parts and people in its history, including its eventual algebraic solution by Carlsson and Mellit, which was published in the Journal of the American Mathematical Society in 2018. Finally, we conclude with some remaining open problems.

Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture

The symmetric group Sn acts on the polynomial ring Q[xn]=Q[x1,…,xn] by variable permutation. The invariant ideal In is the ideal generated by all Sn-invariant polynomials with vanishing constant term. The quotient Rn=Q[xn]In is called the coinvariant algebra. The coinvariant algebra Rn has received a great deal of study in algebraic and geometric combinatorics. We introduce a generalization In,k⊆Q[xn] of the ideal In indexed by two positive integers k≤n. The corresponding quotient Rn,k:=Q[xn]In,k carries a graded action of Sn and specializes to Rn when k=n. We generalize many of the nice properties of Rn to Rn,k. In particular, we describe the Hilbert series of Rn,k, give extensions of the Artin and Garsia–Stanton monomial bases of Rn to Rn,k, determine the reduced Gröbner basis for In,k with respect to the lexicographic monomial order, and describe the graded Frobenius series of Rn,k. Just as the combinatorics of Rn are controlled by permutations in Sn, we will show that the combinatorics of Rn,k are controlled by ordered set partitions of {1,2,…,n} with k blocks. The Delta Conjecture of Haglund, Remmel, and Wilson is a generalization of the Shuffle Conjecture in the theory of diagonal coinvariants. We will show that the graded Frobenius series of Rn,k is (up to a minor twist) the t=0 specialization of the combinatorial side of the Delta Conjecture. It remains an open problem to give a bigraded Sn-module Vn,k whose Frobenius image is even conjecturally equal to any of the expressions in the Delta Conjecture; our module Rn,k solves this problem in the specialization t=0.

The Delta Conjecture

We conjecture two combinatorial interpretations for the symmetric function $\Delta _{e_k} e_n$, where $\Delta _f$ is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture of Haglund, Haiman, Remmel, Loehr, and Ulyanov, which was proved recently by Carlsson and Mellit. We show how previous work of the third author on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.

A proof of the shuffle conjecture

We present a proof of the compositional shuffle conjecture by Haglund, Morse and Zabrocki [Canad. J. Math., 64 (2012), 822–844], which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra by Haglund, Haiman, Loehr, Remmel, and Ulyanov [Duke Math. J., 126 (2005), 195–232]. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector spaceV∗V_*whose degree zero part is the ring of symmetric functionsSym⁡[X]\operatorname {Sym}[X]overQ(q,t)\mathbb {Q}(q,t). We then extend these operators to an action of an algebraA~\tilde {\mathbb A}acting on this space, and interpret the right generalization of the∇\nablausing an involution of the algebra which is antilinear with respect to the conjugation(q,t)↦(q−1,t−1)(q,t)\mapsto (q^{-1},t^{-1}).

A minimaj-preserving crystal on ordered multiset partitions

We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization Rn,k due to Haglund, Rhoades and Shimozono of the coinvariant algebra Rn. The crystal structure also enables us to demonstrate the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.

Ordered set partition statistics and the Delta Conjecture

The Delta Conjecture of Haglund, Remmel, and Wilson is a recent generalization of the Shuffle Conjecture in the field of diagonal harmonics. In this paper we give evidence for the Delta Conjecture by proving a pair of conjectures of Wilson and Haglund–Remmel–Wilson which give equidistribution results for statistics related to inversion count and major index on objects related to ordered set partitions. Our results generalize the famous result of MacMahon that major index and inversion number share the same distribution on permutations.

A proof of the Square Paths Conjecture

The modified Macdonald polynomials, introduced by Garsia and Haiman (1996), have many astounding combinatorial properties. One such class of properties involves applying the related ∇ operator of Bergeron and Garsia (1999) to basic symmetric functions. The first discovery of this type was the (recently proven) Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel, and Ulyanov (2005), which relates the expression ∇en to parking functions. In (2007), Loehr and Warrington conjectured a similar expression for ∇pn which is known as the Square Paths Conjecture.Haglund and Loehr (2005) introduced the notion of schedules to enumerate parking functions with a fixed set of cars in each diagonal. In this paper, we extend the notion of schedules and some related results of Hicks (2013) to labeled square paths. We then apply our new results to prove the Square Paths Conjecture.

A new plethystic symmetric function operator and the rational compositional shuffle conjecture at t = 1/q

Our main result here is that the specialization at t=1/q of the Qkm,kn operators studied in Bergeron et al. [2] may be given a very simple plethystic form. This discovery yields elementary and direct derivations of several identities relating these operators at t=1/q to the Rational Compositional Shuffle conjecture of Bergeron et al. [3]. In particular we show that if m, n and k are positive integers and (m,n) is a coprime pair thenq(km−1)(kn−1)+k−12Qkm,kn(−1)kn|t=1/q=[k]q[km]qekm[X[km]q] where as customarily, for any integer s≥0 and indeterminate u we set [s]u=1+u+⋯+us−1. We also show that the symmetric polynomial on the right hand side is always Schur positive. Moreover, using the Rational Compositional Shuffle conjecture, we derive a precise formula expressing this polynomial in terms of Parking Functions in the km×kn lattice rectangle.

Rational parking functions and LLT polynomials

We prove that the combinatorial side of the “Rational Shuffle Conjecture” provides a Schur-positive symmetric polynomial. Furthermore, we prove that the contribution of a given rational Dyck path can be computed as a certain skew LLT polynomial, thus generalizing the result of Haglund, Haiman, Loehr, Remmel and Ulyanov. The corresponding skew diagram is described explicitly in terms of a certain (m,n)-core.

Rational Parking Functions and Catalan Numbers

The “classical” parking functions, counted by the Cayley number (n+1) n−1, carry a natural permutation representation of the symmetric group S n in which the number of orbits is the Catalan number $${\frac{1}{n+1} \left( \begin{array}{ll} 2n \\ n \end{array} \right)}$$ . In this paper, we will generalize this setup to “rational” parking functions indexed by a pair (a, b) of coprime positive integers. These parking functions, which are counted by b a−1, carry a permutation representation of S a in which the number of orbits is the “rational” Catalan number $${\frac{1}{a+b} \left( \begin{array}{ll} a+b \\ a \end{array} \right)}$$ . First, we compute the Frobenius characteristic of the S a -module of (a, b)-parking functions, giving explicit expansions of this symmetric function in the complete homogeneous basis, the power-sum basis, and the Schur basis. Second, we study q-analogues of the rational Catalan numbers, conjecturing new combinatorial formulas for the rational q-Catalan numbers $${\frac{1}{[a+b]_{q}} {{\left[ \begin{array}{ll} a+b \\ a \end{array} \right]}_{q}}}$$ and for the q-binomial coefficients $${{{\left[ \begin{array}{ll} n \\ k \end{array} \right]}_{q}}}$$ . We give a bijective explanation of the division by [a+b] q that proves the equivalence of these two conjectures. Third, we present combinatorial definitions for q, t-analogues of rational Catalan numbers and parking functions, generalizing the Shuffle Conjecture for the classical case. We present several conjectures regarding the joint symmetry and t = 1/q specializations of these polynomials. An appendix computes these polynomials explicitly for small values of a and b.

Compositional (km,kn)-Shuffle Conjectures

In 2008, Haglund et al. [21] formulated a Compositional form of the Shuffle Conjecture of Haglund et al. [20]. In very recent work, Gorsky and Negut, by combining their discoveries [19, 25, 26] with the work of Schiffmann and Vasserot [28, 29] on the symmetric function side and the work of Hikita [22] and Gorsky and Mazin [18] on the combinatorial side, were led to formulate an infinite family of conjectures that extend the original Shuffle Conjecture of [20]. In fact, they formulated one conjecture for each pair |$(m,n)$| of coprime integers. This work of Gorsky–Negut leads naturally to the question as to where the Compositional Shuffle Conjecture of Haglund–Morse–Zabrocki fits into these recent developments. Our contribution here is a compositional extension of the Gorsky–Negut Shuffle Conjecture for each pair |$(km,kn)$|⁠, with |$(m,n)$| coprime and |$k > 1$|⁠.

Refined knot invariants and Hilbert schemes

We consider the construction of refined Chern–Simons torus knot invariants by M. Aganagic and S. Shakirov from the DAHA viewpoint of I. Cherednik. We give a proof of Cherednik's conjecture on the stabilization of superpolynomials, and then use the results of O. Schiffmann and E. Vasserot to relate knot invariants to the Hilbert scheme of points on C2. Then we use the methods of the second author to compute these invariants explicitly in the uncolored case. We also propose a conjecture relating these constructions to the rational Cherednik algebra, as in the work of the first author, A. Oblomkov, J. Rasmussen and V. Shende. Among the combinatorial consequences of this work is a statement of the mn shuffle conjecture.

A simpler formula for the number of diagonal inversions of an [formula omitted]-parking function and a returning fermionic formula

Recent results have placed the classical shuffle conjecture of Haglund et al. in a broader context of an infinite family of conjectures about parking functions in any rectangular lattice. The combinatorial side of the new conjectures has been defined using a complicated generalization of the dinv statistic that is composed of three parts and that is not obviously non-negative. Here we simplify the definition of dinv, prove that it is always non-negative, and give a graphical description of the statistic in the style of the classical case. We go on to show that in the (n−1)×n lattice, parking functions satisfy a fermionic formula that is similar to the one given in the classical case by Haglund and Loehr.

New combinatorial formulations of the shuffle conjecture

The shuffle conjecture (due to Haglund, Haiman, Loehr, Remmel, and Ulyanov) provides a combinatorial formula for the Frobenius series of the diagonal harmonics module DHn, which is the symmetric function ∇(en). This formula is a sum over all labeled Dyck paths of terms built from combinatorial statistics called area, dinv, and IDes. We provide three new combinatorial formulations of the shuffle conjecture based on other statistics on labeled paths, parking functions, and related objects. Each such reformulation arises by introducing an appropriate new definition of the inverse descent set. Analogous results are proved for the higher-order shuffle conjecture involving ∇m(en). We also give new versions of some recently proposed combinatorial formulas for ∇(Cα) and ∇(s(k,1(n−k))), which translate expansions based on the dinv statistic into equivalent expansions based on Haglund's bounce statistic.