Representations of rational Cherednik algebras of rank one in positive characteristic

In this paper, we classify the irreducible representations of the rational Cherednik algebras of rank 1 in characteristic p>0. There are two cases. One is the “quantum” case, where “Planck's constant” is nonzero and generic irreducible representations have dimension pr, where r is the order of the cyclic group contained in the algebra. The other is the “classical” case, where “Planck's constant” is zero and generic irreducible representations have dimension r.     

Equations resolving a conjecture of Rado on partition regularity

A linear equation L is called k-regular if every k-coloring of the positive integers contains a monochromatic solution to L. Richard Rado conjectured that for every positive integer k, there exists a linear equation that is (k−1)-regular but not k-regular. We prove this conjecture by showing that the equation has this property.This conjecture is part of problem E14 in Richard K. Guy's book “Unsolved Problems in Number Theory”, where it is attributed to Rado's 1933 thesis, “Studien zur Kombinatorik”.     

Waring's number mod m

Let p be an odd prime and γ(k,pn) be the smallest positive integer s such that every integer is a sum of s kth powers . We establish γ(k,pn)?[k/2]+2 and provided that k is not divisible by (p−1)/2. Next, let t=(p−1)/(p−1,k), and q be any positive integer. We show that if ?(t)?q then γ(k,pn)?c(q)k1/q for some constant c(q). These results generalize results known for the case of prime moduli.

Video abstract

For a video summary of this paper, please visit http://www.youtube.com/watch?v=zpHYhwL1kD0.     

Linear recurrence relations for binomial coefficients modulo a prime

We investigate when the sequence of binomial coefficients modulo a prime p, for a fixed positive integer k, satisfies a linear recurrence relation of (positive) degree h in the finite range 0?i?k. In particular, we prove that this cannot occur if 2h?k<p−h. This hypothesis can be weakened to 2h?k<p if we assume, in addition, that the characteristic polynomial of the relation does not have −1 as a root. We apply our results to recover a known bound for the number of points of a Fermat curve over a finite field.     

Finite jet determination of CR mappings

We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M⊂CN, N?2, that is essentially finite and of finite type at each of its points, for every point p∈M there exists an integer ?p, depending upper-semicontinuously on p, such that for every smooth generic submanifold M^′⊂CN of the same dimension as M, if are two germs of smooth finite CR mappings with the same ?p jet at p, then necessarily for all positive integers k. In the hypersurface case, this result provides several new unique jet determination properties for holomorphic mappings at the boundary in the real-analytic case; in particular, it provides the finite jet determination of arbitrary real-analytic CR mappings between real-analytic hypersurfaces in CN of D'Angelo finite type. It also yields a new boundary version of H. Cartan's uniqueness theorem: if Ω,Ω^′⊂CN are two bounded domains with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary ∂Ω, such that if are two proper holomorphic mappings extending smoothly up to ∂Ω near some point p∈∂Ω and agreeing up to order k at p, then necessarily H₁=H₂.     

Simple arguments on consecutive power residues

By some extremely simple arguments, we point out the following:

(i)

If n is the least positive kth power non-residue modulo a positive integer m, then the greatest number of consecutive kth power residues mod m is smaller than m/n.

(ii)

Let OK be the ring of algebraic integers in a quadratic field with d∈{−1,−2,−3,−7,−11}. Then, for any irreducible π∈OK and positive integer k not relatively prime to , there exists a kth power non-residue ω∈OK modulo π such that .

     

Galois deformation theory for norm fields and flat deformation rings

Let K be a finite extension of Qp, and choose a uniformizer π∈K, and put . We introduce a new technique using restriction to to study flat deformation rings. We show the existence of deformation rings for -representations “of height ≤h” for any positive integer h, and prove that when h=1 they are isomorphic to “flat deformation rings”. This -deformation theory has a good positive characteristics analogue of crystalline representations in the sense of Genestier-Lafforgue. In particular, we obtain a positive characteristic analogue of crystalline deformation rings, and can analyze their local structure.     

Integrals over Grassmannians and random permutations

Testing the independence of two Gaussian populations involves the distribution of the sample canonical correlation coefficients, given that the actual correlation is zero. The “Laplace transform” (as a function of x) of this distribution is not only an integral over the Grassmannian of p-dimensional planes in real, complex or quaternion n-space , but is also related to a generalized hypergeometric function. Such integrals are solutions of Painlevé-like equations; in the complex case, they are solutions to genuine Painlevé equations. These integrals over have remarkable expansions in x, related to random words of length ? formed with an alphabet of p letters 1,…,p. The coefficients of these expansions are given by the probability that a word (i) contains a subsequence of letters p,p−1,…,1 in that order and (ii) that the maximal length of the disjoint union of p−1 increasing subsequences of the word is ?k, where k refers to the power of x. Note that, if each letter appears in the word, then the maximal length of the disjoint union of p increasing subsequences of the word is automatically =? and is thus trivial.     

Remarks on the minus (signed) total domination in graphs

A function f:V(G)→{+1,0,-1} defined on the vertices of a graph G is a minus total dominating function if the sum of its function values over any open neighborhood is at least 1. The minus total domination number of G is the minimum weight of a minus total dominating function on G. By simply changing “{+1,0,-1}” in the above definition to “{+1,-1}”, we can define the signed total dominating function and the signed total domination number of G. In this paper we present a sharp lower bound on the signed total domination number for a k-partite graph, which results in a short proof of a result due to Kang et al. on the minus total domination number for a k-partite graph. We also give sharp lower bounds on and for triangle-free graphs and characterize the extremal graphs achieving these bounds.     

Ono invariants of imaginary quadratic fields with class number three

Let Ed(x) denote the “Euler polynomial” x²+x+(1−d)/4 if and x²−d if . Set Ω(n) to be the number of prime factors (counting multiplicity) of the positive integer n. The Ono invariantOnod of is defined to be except when d=−1,−3 in which case Onod is defined to be 1. Finally, let hd=hk denote the class number of K. In 2002 J. Cohen and J. Sonn conjectured that hd=3⇔Onod=3 and is a prime. They verified that the conjecture is true for p<1.5×¹⁰⁷. Moreover, they proved that the conjecture holds for p>¹⁰¹⁷ assuming the extended Riemann Hypothesis. In this paper, we show that the conjecture holds for p?2.5×¹⁰¹³ by the aid of computer. And using a result of Bach, we also proved that the conjecture holds for p>2.5×¹⁰¹³ assuming the extended Riemann Hypothesis. In conclusion, we proved the conjecture is true assuming the extended Riemann Hypothesis.     

Composite of irreducible morphisms in the bounded derived category

We study necessary and sufficient conditions for the existence of n irreducible morphisms in the bounded derived category of an Artin algebra, with non-zero composite in the n+1-power of the radical. In the case of , the bounded derived category of an Ext-finite hereditary k-category with tilting object, such irreducible morphisms exist if and only if H is derived equivalent to a wild hereditary algebra or to a wild canonical algebra. We also characterize the cluster tilted algebras having such irreducible morphisms.     

Diameters of uniform subset graphs

Let r,s be positive integers with r>s, k a nonnegative integer, and n=2r−s+k. A uniform subset graph G(n,r,s) is a graph with vertex set [n]^r and where two r-subsets A,B∈[n]^r are adjacent if and only if |A∩B|=s. Let denote the diameter of a graph G.In this paper, we prove the following results: (1) If k>0, then if r≥2s+k+2, 2 if k≥s and 2s≤r≤s+k, or k<s and s+k≤r≤2s, and 3 otherwise; (2) If k=0, then . This generalizes a result in [M. Valencia-Pabon, J.-C. Vera, On the diameter of Kneser graphs, Discrete Math. 305 (2005) 383-385].     

Equitable colorings of Kronecker products of graphs

For a positive integer k, a graph G is equitably k-colorable if there is a mapping f:V(G)→{1,2,…,k} such that f(x)≠f(y) whenever xy∈E(G) and ||f⁻¹(i)|−|f⁻¹(j)||≤1 for 1≤i<j≤k. The equitable chromatic number of a graph G, denoted by χ₌(G), is the minimum k such that G is equitably k-colorable. The equitable chromatic threshold of a graph G, denoted by , is the minimum t such that G is equitably k-colorable for k≥t. The current paper studies equitable chromatic numbers of Kronecker products of graphs. In particular, we give exact values or upper bounds on χ₌(G×H) and when G and H are complete graphs, bipartite graphs, paths or cycles.     

The unramified computation of Rankin–Selberg integrals expressed in terms of Bessel models for split orthogonal groups: Part I

In this paper, we compute the local integrals, with normalized unramified data, over a p-adic field F, arising from general Rankin–Selberg integrals for SO _m × GL_r+k+1, where the orthogonal group is split over F, \(k \leqslant \left[ {\frac{{m - 1}}{2}} \right]\), and the irreducible representation of SO _m (F) has a Bessel model with respect to an irreducible representation of the split orthogonal group SO_m?2k?1(F). Our proof is by “analytic continuation from the unramified computation in the generic case”. We let the unramified parameters of the representations involved vary, and express the local integrals in terms of the Whittaker models of the representations, which exist at points in general position. Then we apply analytic continuation and the known unramified computation in the generic case. We discuss some applications to poles of partial L-functions and functorial lifting.     

Linear pencils on real algebraic curves

Our knowledge of linear series on real algebraic curves is still very incomplete. In this paper we restrict to pencils (complete linear series of dimension one). Let X denote a real curve of genus g with real points and let k(R) be the smallest degree of a pencil on X (the real gonality of X). Then we can find on X a base point free pencil of degree g+1 (resp. g if X is not hyperelliptic, i.e. if k(R)>2) with an assigned geometric behaviour w.r.t. the real components of X, and if we prove that which is the same bound as for the gonality of a complex curve of even genus g. Furthermore, if the complexification of X is a k-gonal curve (k≥2) one knows that k≤k(R)≤2k−2, and we show that for any two integers k≥2 and 0≤n≤k−2 there is a real curve with real points and k-gonal complexification such that its real gonality is k+n.     

EKR type inequalities for 4-wise intersecting families

Let 1?t?7 be an integer and let F be a k-uniform hypergraph on n vertices. Suppose that |A∩B∩C∩D|?t holds for all A,B,C,D∈F. Then we have if holds for some ε>0 and all n>n₀(ε). We apply this result to get EKR type inequalities for “intersecting and union families” and “intersecting Sperner families.”     

A multiply intersecting Erd?s-Ko-Rado theorem — The principal case

Let m(n,k,r,t) be the maximum size of satisfying |F₁∩?∩F_r|≥t for all F₁,…,F_r∈F. We prove that for every p∈(0,1) there is some r₀ such that, for all r>r₀ and all t with 1≤t≤⌊(p^1−r−p)/(1−p)⌋−r, there exists n₀ so that if n>n₀ and p=k/n, then . The upper bound for t is tight for fixed p and r.     

Colouring graphs with bounded generalized colouring number

Given a graph G and a positive integer p, χ_p(G) is the minimum number of colours needed to colour the vertices of G so that for any i≤p, any subgraph H of G of tree-depth i gets at least i colours. This paper proves an upper bound for χ_p(G) in terms of the k-colouring number of G for k=2^p−2. Conversely, for each integer k, we also prove an upper bound for in terms of χ_k+2(G). As a consequence, for a class K of graphs, the following two statements are equivalent:

(a)

For every positive integer p, χ_p(G) is bounded by a constant for all G∈K.

(b)

For every positive integer k, is bounded by a constant for all G∈K.

It was proved by Nešet?il and Ossona de Mendez that (a) is equivalent to the following:

(c)

For every positive integer q, ∇_q(G) (the greatest reduced average density of G with rank q) is bounded by a constant for all G∈K.

Therefore (b) and (c) are also equivalent. We shall give a direct proof of this equivalence, by introducing ∇_q−(1/2)(G) and by showing that there is a function F_k such that . This gives an alternate proof of the equivalence of (a) and (c).     

Polynomial Cunningham chains

A sequence of prime numbers p₁,p₂,p₃,…, such that pi=2pi−1+? for all i, is called a Cunningham chain of the first or second kind, depending on whether ?=1 or −1 respectively. If k is the smallest positive integer such that 2pk+? is composite, then we say the chain has length k. It is conjectured that there are infinitely many Cunningham chains of length k for every positive integer k. A sequence of polynomials f₁(x),f₂(x),… in Z[x], such that f₁(x) has positive leading coefficient, each fi(x) is irreducible in Q[x] and fi(x)=xfi−1(x)+? for all i, is defined to be a polynomial Cunningham chain of the first or second kind, depending on whether ?=1 or −1 respectively. If k is the least positive integer such that fk+1(x) is reducible in Q[x], then we say the chain has length k. In this article, for polynomial Cunningham chains of both kinds, we prove that there are infinitely many chains of length k and, unlike the situation in the integers, that there are infinitely many chains of infinite length, by explicitly giving infinitely many polynomials f₁(x), such that fk+1(x) is the only term in the sequence that is reducible.