Constant Term Identities Research Articles

The AFLT q-Morris constant term identity

It is well-known that the Selberg integral is equivalent to the Morris constant term identity. More generally, Selberg type integrals can be turned into constant term identities for Laurent polynomials. In this paper, by extending the Gessel–Xin method of the Laurent series proof of constant term identities, we obtain an AFLT type q-Morris constant term identity. That is a q-Morris type constant term identity for a product of two Macdonald polynomials.

Two families of constant term identities

In 1985, Bressoud and Goulden derived the formula for the constant term in $$\prod _{(i,j)\in T} \frac{x_j}{x_i} \prod _{0\le i<j \le n}(\frac{x_i}{x_j})_{a_i}(\frac{qx_j}{x_i})_{a_j-1}$$ , where $$T \subseteq \{(i,j)\mid 0\le i<j \le n\}$$ . This result implies the Andrews’ q-Dyson identity. In 2006, Gessel and Xin proved the q-Dyson identity by considering both sides of the equality as polynomials in $$q^{a_0}$$ . We use this approach to determine the coefficients of $$x_0/x_1$$ and $$x_0/x_2$$ in Laurent polynomials studied by Bressoud and Goulden.

Refinements and Symmetries of the Morris identity for volumes of flow polytopes

Flow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations. The Chan-Robbins-Yuen (CRY) polytope is a flow polytope with normalized volume equal to the product of consecutive Catalan numbers. Zeilberger proved this by evaluating the Morris constant term identity, but no combinatorial proof is known. There is a refinement of this formula that splits the largest Catalan number into Narayana numbers, which M\'esz\'aros gave an interpretation as the volume of a collection of flow polytopes. We introduce a new refinement of the Morris identity with combinatorial interpretations both in terms of lattice points and volumes of flow polytopes. Our results generalize M\'esz\'aros's construction and a recent flow polytope interpretation of the Morris identity by Corteel-Kim-M\'esz\'aros. We prove the product formula of our refinement following the strategy of the Baldoni-Vergne proof of the Morris identity. Lastly, we study a symmetry of the Morris identity bijectively using the Danilov-Karzanov-Koshevoy triangulation of flow polytopes and a bijection of M\'esz\'aros-Morales-Striker.

On the q-Dyson orthogonality problem

By combining the Gessel–Xin method with plethystic substitutions, we obtain a recursion for a symmetric function generalization of the q-Dyson constant term identity also known as the Zeilberger–Bressoud q-Dyson theorem. This yields a constant term identity which generalizes the non-zero part of Kadell's orthogonality ex-conjecture and a result of Károlyi, Lascoux and Warnaar.

A recursion for a symmetric function generalization of the q-Dyson constant term identity

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the q-Dyson constant term identity or the Zeilberger–Bressoud q-Dyson theorem. The non-zero part of Kadell's orthogonality conjecture is a constant term identity indexed by a weak composition v=(v1,…,vn) in the case when only one vi≠0. This conjecture was first proved by Károlyi, Lascoux and Warnaar in 2015. They further formulated a closed-form expression for the above mentioned constant term in the case when all the parts of v are distinct. Recently we obtained a recursion for this constant term provided that the largest part of v occurs with multiplicity one in v. In this paper, we generalize our previous result to all weak compositions v.

Volumes of Flow Polytopes Related to Caracol Graphs

Recently, Benedetti et al. introduced an Ehrhart-like polynomial associated to a graph. This polynomial is defined as the volume of a certain flow polytope related to a graph and has the property that the leading coefficient is the volume of the flow polytope of the original graph with net flow vector $(1,1,\dots,1)$. Benedetti et al. conjectured a formula for the Ehrhart-like polynomial of what they call a caracol graph. In this paper their conjecture is proved using constant term identities, labeled Dyck paths, and a cyclic lemma.

A symmetric function generalization of the Zeilberger–Bressoud $q$-Dyson theorem

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the Zeilberger--Bressoud $q$-Dyson theorem or the $q$-Dyson constant term identity. This conjecture was proved by Karolyi, Lascoux and Warnaar in 2015. In this paper, by slightly changing the variables of Kadell's conjecture, we obtain another symmetric function generalization of the $q$-Dyson constant term identity. This new generalized constant term admits a simple product-form expression.

Quantum Q-Systems and Fermionic Sums—The Non-Simply Laced Case

Abstract In this paper, we seek to prove the equality of the $q$-graded fermionic sums conjectured by Hatayama et al. [ 14] in its full generality, by extending the results of Di Francesco and Kedem [ 9] to the non-simply laced case. To this end, we will derive explicit expressions for the quantum $Q$-system relations, which are quantum cluster mutations that correspond to the classical $Q$-system relations, and write the identity of the $q$-graded fermionic sums as a constant term identity. As an application, we will show that these quantum $Q$-system relations are consistent with the short exact sequence of the Feigin–Loktev fusion product of Kirillov–Reshetikhin modules obtained by Chari and Venkatesh [ 5].

Refined enumerations of alternating sign triangles

This article introduces and investigates a refinement of alternating sign trapezoids by means of Catalan objects and Motzkin paths. Alternating sign trapezoids are a generalisation of alternating sign triangles, which were recently introduced by Ayyer, Behrend and Fischer. We show that the number of alternating sign trapezoids associated with a Catalan object (resp. a Motzkin path) is a polynomial function in the length of the shorter base of the trapezoid. We also study the rational roots of these polynomials and formulate several conjectures and derive some partial results. Lastly, we deduce a constant term identity for the refined counting of alternating sign trapezoids.

Flow polytopes with Catalan volumes

The Chan–Robbins–Yuen polytope can be thought of as the flow polytope of the complete graph with netflow vector (1,0,…,0,−1). The normalized volume of the Chan–Robbins–Yuen polytope equals the product of consecutive Catalan numbers, yet there is no combinatorial proof of this fact. We consider a natural generalization of this polytope, namely, the flow polytope of the complete graph with netflow vector (1,1,0,…,0,−2). We show that the volume of this polytope is a certain power of 2 times the product of consecutive Catalan numbers. Our proof uses constant-term identities and further deepens the combinatorial mystery of why these numbers appear. In addition, we introduce two more families of flow polytopes whose volumes are given by product formulas.

Leading coefficients of Morris type constant term identities

The Morris constant term identity is known to be equivalent to the famous Selberg integral. In this paper, we regard Morris type constant terms as polynomials of a certain parameter. Thus, we can take the leading coefficients and obtain new identities. These identities happen to be crucial in finding lower bounds for cardinalities of some restricted sumsets, and in calculating the volume of a Tesler polytope.

Twisted strong Macdonald theorems and adjoint orbits

The strong Macdonald theorems state that, for L reductive and s an odd variable, the cohomology algebras H⁎(L[z]/zN) and H⁎(L[z,s]) are freely generated, and describe the cohomological, s-, and z-degrees of the generators. The resulting identity for the z-weighted Euler characteristic is equivalent to Macdonald's constant term identity for a finite root system. We calculate H⁎(p/zNp) and H⁎(p[s]) for p a standard parahoric in a twisted loop algebra, giving strong Macdonald theorems that take into account both a parabolic component and a possible diagram automorphism twist. In particular we show that H⁎(p/zNp) contains a parabolic subalgebra of the coinvariant algebra of the fixed-point subgroup of the Weyl group of L, and thus is no longer free. We also prove a strong Macdonald theorem for H⁎(b;S⁎n⁎) and H⁎(b/zNn) when b and n are Iwahori and nilpotent subalgebras respectively of a twisted loop algebra. For each strong Macdonald theorem proved, taking z-weighted Euler characteristics gives an identity equivalent to Macdonald's constant term identity for the corresponding affine root system. As part of the proof, we study the regular adjoint orbits for the adjoint action of the twisted arc group associated to L, proving an analogue of the Kostant slice theorem.

Constant term identities and Poincaré polynomials

In 1982 Macdonald published his now famous constant term conjectures for classical root systems. This paper begins with the almost trivial observation that Macdonald’s constant term identities admit an extra set of free parameters, thereby linking them to Poincaré polynomials. We then exploit these extra degrees of freedom in the case of typeA\mathrm {A}to give the first proof of Kadell’s orthogonality conjecture—a symmetric function generalisation of theqq-Dyson conjecture or Zeilberger–Bressoud theorem.Key ingredients in our proof of Kadell’s orthogonality conjecture are multivariable Lagrange interpolation, the scalar product for Demazure characters and(0,1)(0,1)-matrices.

A new approach to constant term identities and Selberg-type integrals

Selberg-type integrals that can be turned into constant term identities for Laurent polynomials arise naturally in conjunction with random matrix models in statistical mechanics. Built on a recent idea of Karasev and Petrov we develop a general interpolation based method that is powerful enough to establish many such identities in a simple manner. The main consequence is the proof of a conjecture of Forrester related to the Calogero–Sutherland model. In fact we prove a more general theorem, which includes Aomoto's constant term identity at the same time. We also demonstrate the relevance of the method in additive combinatorics.

Vertex Algebras W(p)Amand W(p)Dmand Constant Term Identities

We consider $AD$-type orbifolds of the triplet vertex algebras $\mathcal{W}(p)$ extending the well-known $c=1$ orbifolds of lattice vertex algebras. We study the structure of Zhu's algebras $A(\mathcal{W}(p)^{A_m})$ and $A(\mathcal{W}(p)^{D_m})$, where $A_m$ and $D_m$ are cyclic and dihedral groups, respectively. A combinatorial algorithm for classification of irreducible $\mathcal{W}(p)^\Gamma$-modules is developed, which relies on a family of constant term identities and properties of certain polynomials based on constant terms. All these properties can be checked for small values of $m$ and $p$ with a computer software. As a result, we argue that if certain constant term properties hold, the irreducible modules constructed in [Commun. Contemp. Math. 15 (2013), 1350028, 30 pages, arXiv:1212.5453; Internat. J. Math. 25 (2014), 1450001, 34 pages, arXiv:1304.5711] provide a complete list of irreducible $\mathcal{W}(p)^{A_m}$ and $\mathcal{W}(p)^{D_m}$-modules. This paper is a continuation of our previous work on the $ADE$ subalgebras of the triplet vertex algebra $\mathcal{W}(p)$.

ADESubalgebras of the Triplet Vertex Algebra 𝒲(p):E6,E7

It is the forth paper in a series devoted to the investigation of the |$ADE$| subalgebras of the triplet vertex algebra |$\mathcal {W}(p)$|⁠. In this paper, we prove |$C_{2}$|-cofiniteness of the invariant subalgebras |$\mathcal {W}(p)^{\Gamma }$| and study their representations when |$\Gamma$| is the tetrahedral group |$T$| (of |$E_6$|-type), or the octahedral group |$O$| (of |$E_7$|-type). We give a list of |$21p$| irreducible |$\mathcal {W}(p)^{T}$|-modules and a list of |$28p$| irreducible |$\mathcal {W}(p)^{O}$|-modules. The completeness of these two lists can be reduced to some conjectures about constant term identities which have been verified in the case |$p=1$|⁠. For the proof of |$C_{2}$|-cofiniteness of |$\mathcal {W}(p)^{\Gamma }$|⁠, and the calculations of the Zhu's algebras |$A(\mathcal {W}(p)^{\Gamma })$|⁠, we develop several quite universal formulas (Theorem 4.3 and its corollaries). In Appendix, we further apply them to derive a useful tool of structure coefficients for the triplet vertex algebra |$\mathcal {W}(p)$| and its irreducible modules, which unifies almost all the calculations in this paper; as an unexpected byproduct, we give a new proof of |$\mathfrak {sl}_{2}$|-action on |$\mathcal {W}(p)$|⁠.

A Laurent Series Proof of the Habsieger-Kadell $q$-Morris Identity

We give a Laurent series proof of the Habsieger-Kadell $q$-Morris identity, which is a common generalization of the $q$-Morris identity and the Aomoto constant term identity. The proof allows us to extend the theorem for some additional parameter cases.

Connectivity Patterns in Loop Percolation I: the Rationality Phenomenon and Constant Term Identities

Loop percolation, also known as the dense O(1) loop model, is a variant of critical bond percolation in the square lattice Z^2 whose graph structure consists of a disjoint union of cycles. We study its connectivity pattern, which is a random noncrossing matching associated with a loop percolation configuration. These connectivity patterns exhibit a striking rationality property whereby probabilities of naturally-occurring events are dyadic rational numbers or rational functions of a size parameter n, but the reasons for this are not completely understood. We prove the rationality phenomenon in a few cases and prove an explicit formula expressing the probabilities in the "cylindrical geometry" as coefficients in certain multivariate polynomials. This reduces the rationality problem in the general case to that of proving a family of conjectural constant term identities generalizing an identity due to Di Francesco and Zinn-Justin. Our results make use of, and extend, algebraic techniques related to the quantum Knizhnik-Zamolodchikov equation.

Wronskian type determinants of orthogonal polynomials, Selberg type formulas and constant term identities

Let (pn)n be a sequence of orthogonal polynomials with respect to the measure μ. Let T be a linear operator acting in the linear space of polynomials P and satisfying deg(T(p))=deg(p)−1, for all polynomial p. We then construct a sequence of polynomials (sn)n, depending on T but not on μ, such that the Wronskian type n×n determinant det(Ti−1(pm+j−1(x)))i,j=1n is equal to the m×m determinant det(qn+i−1j−1(x))i,j=1m, up to multiplicative constants, where the polynomials qni, n,i⩾0, are defined by qni(x)=∑j=0nμjisn−j(x), and μji are certain generalized moments of the measure μ. For T=d/dx we recover a theorem by Leclerc which extends the well-known Karlin and Szegő identities for Hankel determinants whose entries are ultraspherical, Laguerre and Hermite polynomials. For T=Δ, the first order difference operator, we get some very elegant symmetries for Casorati determinants of classical discrete orthogonal polynomials. We also show that for certain operators T, the second determinant above can be rewritten in terms of Selberg type integrals, and that for certain operators T and certain families of orthogonal polynomials (pn)n, one (or both) of these determinants can also be rewritten as the constant term of certain multivariate Laurent expansions.

ADE SUBALGEBRAS OF THE TRIPLET VERTEX ALGEBRA 𝒲(p): D-SERIES

We are continuing our study of ADE-orbifold subalgebras of the triplet vertex algebra 𝒲(p) initiated in D. Adamović, X. Lin and A. Milas, ADE subalgebras of the triplet vertex algebra 𝒲(p): Am-series, Commun. Contemp. Math.15 (2013), Article ID: 1350028, 1–30. This part deals with the dihedral series. First, subject to a certain constant term identity, we classify all irreducible modules for the vertex algebra [Formula: see text], the ℤ2-orbifold of the singlet vertex algebra [Formula: see text]. Then, we classify irreducible modules and determine Zhu's and C2-algebra for the vertex algebra 𝒲(p)D2. A general method for construction of twisted 𝒲(p)-modules is also introduced. We also discuss classification of twisted [Formula: see text]-modules including the twisted Zhu's algebra [Formula: see text], which is of independent interest. The category of admissible Ψ-twisted [Formula: see text]-modules is expected to be semisimple. We also prove C2-cofiniteness of 𝒲(p)Dm for all m, and give a conjectural list of irreducible 𝒲(p)Dm-modules. Finally, we compute characters of the relevant irreducible modules and describe their modular closure.