Monodromies at infinity of confluent A-hypergeometric functions

We study the monodromies at infinity of confluent A-hypergeometric functions introduced by Adolphson [2]. In particular, we extend the result of [38] for non-confluent A-hypergeometric functions to the confluent case. The integral representation by rapid decay homology cycles proved in [9] will play a central role in the proof.     

Dualizing,projecting, and restricting GKZ systems

Let A be an integer matrix, and assume that its semigroup ring $\mathbb{C}[\mathbb{N}A]$ is normal. Fix a face F of the cone of A. We show that the projection and restriction of an A-hypergeometric system to the coordinate subspace corresponding to F are essentially F-hypergeometric; moreover, at most one of them is nonzero.We also show that, if A is in addition homogeneous, the holonomic dual of an A-hypergeometric system is itself A-hypergeometric. This extends a result from [16], proving a conjecture of Nobuki Takayama in the normal homogeneous case.     

Planar Configurations of Lattice Vectors and GKZ-Rational Toric Fourfolds in ℙ6

We introduce a notion of balanced configurations of vectors. This is motivated by the study of rational A-hypergeometric functions in the sense of Gelfand, Kapranov and Zelevinsky. We classify balanced configurations of seven plane vectors up to GL(2,)-equivalence and deduce that the only gkz-rational toric four-folds in ⁶ are those varieties associated with an essential Cayley configuration. We show that in this case, all rational A-hypergeometric functions may be described in terms of toric residues. This follows from studying a suitable hyperplane arrangement.     

Zonal spherical functions on the complex reflection groups and (n+1,m+1)-hypergeometric functions

The pair of groups, complex reflection group G(r,1,n) and symmetric group S_n, is a Gelfand pair. Its zonal spherical functions are expressed in terms of multivariate hypergeometric functions called (n+1,m+1)-hypergeometric functions. Since the zonal spherical functions have orthogonality, they form discrete orthogonal polynomials. Also shown is a relation between monomial symmetric functions and the (n+1,m+1)-hypergeometric functions.     

Algebraic A-hypergeometric functions

We formulate and prove a combinatorial criterion to decide if an A-hypergeometric system of differential equations has a full set of algebraic solutions or not. This criterion generalises the so-called interlacing criterion in the case of hypergeometric functions of one variable.     

Partial theta functions and mock modular forms as q-hypergeometric series

Ramanujan studied the analytic properties of many q-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious q-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have q-expansions resembling modular theta functions, is not well understood. Here we consider families of q-hypergeometric series which converge in two disjoint domains. In one domain, we show that these series are often equal to one another, and define mock theta functions, including the classical mock theta functions of Ramanujan, as well as certain combinatorial generating functions, as special cases. In the other domain, we prove that these series are typically not equal to one another, but instead are related by partial theta functions.     

A-hypergeometric series and the Hasse–Witt matrix of a hypersurface

We give a short combinatorial proof of the generic invertibility of the Hasse–Witt matrix of a projective hypersurface. We also examine the relationship between the Hasse–Witt matrix and certain A-hypergeometric series, which is what motivated the proof.     

Cohen-Macaulayness and computation of Newton graded toric rings

Let H⊆Z^d be a positive semigroup generated by A⊆H, and let K[H] be the associated semigroup ring over a field K. We investigate heredity of the Cohen-Macaulay property from K[H] to both its A-Newton graded ring and to its face rings. We show by example that neither one inherits in general the Cohen-Macaulay property. On the positive side, we show that for every H there exist generating sets A for which the Newton graduation preserves Cohen-Macaulayness. This gives an elementary proof for an important vanishing result on A-hypergeometric Euler-Koszul homology. As a tool for our investigations we develop an algorithm to compute algorithmically the Newton filtration on a toric ring.     

On properties and applications of (p,q)-extended τ-hypergeometric functions

We introduce the $(p,q)$-extended τ-hypergeometric and confluent hypergeometric functions along with their integral representations. We also present closed integral expressions for the Mathieu-type a-series and for the associated alternating versions whose terms contain the $(p,q)$-extended τ-hypergeometric functions with related contiguous functional relations.     

Mellin Transforms of Multivariate Rational Functions

This paper deals with Mellin transforms of rational functions g/f in several variables. We prove that the polar set of such a Mellin transform consists of finitely many families of parallel hyperplanes, with all planes in each such family being integral translates of a specific facial hyperplane of the Newton polytope of the denominator f. The Mellin transform is naturally related to the so-called coamoeba $\mathcal{A}'_{f}:=\mathrm{Arg}(Z_{f})$ , where Z _f is the zero locus of f and Arg denotes the mapping that takes each coordinate to its argument. In fact, each connected component of the complement of the coamoeba $\mathcal{A}'_{f}$ gives rise to a different Mellin transform. The dependence of the Mellin transform on the coefficients of f, and the relation to the theory of A-hypergeometric functions is also discussed in the paper.     

One-Dimensional Nil-Daha and Whittaker Functions I

This work is devoted to the theory of nil-DAHA for the root system A ₁ and its applications to symmetric and nonsymmetric (spinor) global q-Whittaker functions, integrating the q-Toda eigenvalue problem and its Dunkl-type nonsymmetric version. The global symmetric function can be interpreted as the generating function of the Demazure characters for dominant weights, which describe the algebraic-geometric properties of the corresponding affine Schubert varieties. Its Harish-Chandra-type asymptotic expansion appears directly related to the solution of the q-Toda eigenvalue problem obtained by Givental and Lee in the quantum K-theory of ag varieties. It provides an exact mathematical relation between the corresponding physics A-type and B-type models. The spinor global functions extend the symmetric ones to the case of all Demazure characters (not only those for dominant weights); the corresponding Gromov?Witten theory is not known. The main result of this work is a complete algebraic theory of these functions in terms of induced modules of the core subalgebra of nil-DAHA. It is the first instance of the DAHA theory of canonical-crystal bases, quite non-trivial even for A ₁. As the first part of the work, this paper is devoted mainly to the analytic aspects of our construction and the beginning of a systematic algebraic theory of nil-DAHA; the second part will be about the induced modules and their applications to the nonsymmetric global Whittaker functions.     

Families of multisums as mock theta functions

In view of the Bailey lemma and the relations between Hecke-type sums and Appell–Lerch sums given by Hickerson and Mortenson, we find that many Bailey pairs given by Slater can be used to deduce mock theta functions. Therefore, by constructing generalized Bailey pairs with more parameters, we derive some new families of mock theta functions. Meanwhile, some identities between new mock theta functions and classical ones are established. Furthermore, based on the proofs of the main theorems, many q-hypergeometric transformations are obtained.     

On quiver varieties and affine Grassmannians of type A

We construct Nakajima's quiver varieties of type A in terms of affine Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. Consequently, singularities of quiver varieties, nilpotent orbits and affine Grassmannians are the same in type A. The construction also provides a geometric framework for skew (GL(m),GL(n)) duality and identifies the natural basis of weight spaces in Nakajima's construction with the natural basis of multiplicity spaces in tensor products which arises from affine Grassmannians. To cite this article: I. Mirkovi?, M. Vybornov, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Isomorphism Classes of A-Hypergeometric Systems

Given a finite set A of integral vectors and a parameter vector, Gel'fand, Kapranov, and Zelevinskii defined a system of differential equations, called an A-hypergeometric (or a GKZ hypergeometric) system. Classifying the parameters according to the D-isomorphism classes of their corresponding A-hypergeometric systems is one of the most fundamental problems in the theory. In this paper we give a combinatorial answer for the problem under the assumption that the finite set A lies in a hyperplane off the origin, and illustrate it in two particularly simple cases: the normal case and the monomial curve case.     

Duality and monodromy reducibility of <Emphasis Type="Italic">A</Emphasis>-hypergeometric systems

We study hypergeometric systems H _A (β) in the sense of Gelfand, Kapranov and Zelevinsky under two aspects: the structure of their holonomically dual system, and reducibility of their rank module. We prove in the first part that rank-jumping parameters always correspond to reducible systems. We show further that the property of being reducible is “invariant modulo the lattice”, and obtain as a special instance a theorem of Alicia Dickenstein and Timur Sadykov on reducibility of Mellin systems. In the second part we study a conjecture of Nobuki Takayama which states that the holonomic dual of H _A (β) is of the form H _A (β′) for suitable β′. We prove the conjecture for all matrices A and generic parameter β, exhibit an example that shows that in general the conjecture cannot hold, and present a refined version of the conjecture. Questions on both duality and reducibility have been quite difficult to answer with classical methods. This paper may be seen as an example of the usefulness, and scope of applications, of the homological tools for A-hypergeometric systems developed in Matusevich et al. (J. Amer. Math. Soc. 18:919–941, 2005)     

Algebras generated by inner functions

Let A be a uniform algebra on a compact space X. An inner function is a function in A unimodular on X. For three algebras of type H^∞ we prove A is generated by its inner functions. Whenever A is generated by its inner functions we prove the unit ball of A is the closed convex ball of the inner functions.     

New identities involving sums of the tails related to real quadratic fields

In previous work, the authors discovered new examples of q-hypergeometric series related to the arithmetic of $\mathbb {Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ . Building on this work, we construct in this paper sum of the tails identities for which some which some of these functions occur as error terms. As an application, we obtain formulas for the generating function of a certain zeta functions for real quadratic fields at negative integers.     

Finite generation of lifted p-adic homology with compact supports. Generalization of the weil conjectures to singular,non-complete algebraic varieties

In Section 1, if $\text{O}$ is a c.d.v.r. with quotient field of characteristic zero and residue class field k, if $\text{A}$ is an $\text{O}$-algebra and if $\text{A =}\text{A}$ ?_$\text{O}$k, then for algebraic families X over A that are polynomially properly embeddable over $\text{A}$, we define the lifted p-adic homology with compact supports$\text{H}{}_{\mathrm{h}}{}^{\mathrm{c}}\text{(X,}\text{A}$2 ?_zQ), which are functors with respect to proper maps. In Section 2, it is shown that, if X is an algebraic variety over k (i.e., if A = k), then the lifted p-adic homology of X with compact supports with coefficients in K is finite dimensional over K = quotient field of $\text{O}$. In Section 3, the results of Sections 1 and 2 are used to generalize both the statement and proof of the Weil “Lefschetz Theorem” Conjecture and the statement (but not the proof) of the Weil “Riemann Hypothesis” Conjecture, to non-complete, singular varieties over finite fields. In addition, the Weil zeta function of varieties over finite fields, is generalized by a device which we call the zeta matrices, W^h(X), 0 ≤ h ≤ 2 dim X, of an algebraic variety X, to varieties over even infinite fields of non-zero characteristic. These are used to give formulas for the zeta functions of each variety in an algebraic family, by means of the zeta matrices of an alebraic family. Sketches only are given. In Section 4, some of the material is duplicated, to define a q-adic homology with compact supports, q ≠ characteristic. The definition only makes sense for algebraic varieties; finite generation is proved. And the Weil “Lefschetz Theorem” Conjecture is established, even for singular, non-complete varieties, as well as a generalization of the Weil “Riemann Hypothesis” Conjecture. (However, zeta matrices do not make sense q-adically.)In Section 5, some special results are proved about p-adic homology with compact supports on affines. And the Weil “Riemann Hypothesis” conjecture is proved p-adically, p = characteristic, for projective, non-singular liftable varieties.     

Topologically inseparable functions I: Finitary case

Given a finite set A and a distinguished function f: A → A, we study the set of all functions g: A → A that are continuous for all topologies for which f is continuous. The main result is a characterization of the functions f such that this set is trivial, that is, contains only the constant functions and the iterates of f.