arXiv subjects

Christian Krattenthaler

Publications and source records attributed to Christian Krattenthaler.

At least 91 records · Page 5Linked to original sources

arXiv2010

Stanley decompositions and Hilbert depth in the Koszul complex

Stanley decompositions of multigraded modules $M$ over polynomials rings have been discussed intensively in recent years. There is a natural notion of depth that goes with a Stanley decomposition, called the Stanley depth. Stanley conjectured that the Stanley depth of a module $M$ is always at least the (classical) depth of $M$. In this paper we introduce a weaker type of decomposition, which we call Hilbert decomposition, since it only depends on the Hilbert function of $M$, and an analogous notion of depth, called Hilbert depth. Since Stanley decompositions are Hilbert decompositions, the latter set upper bounds to the existence of Stanley decompositions. The advantage of Hilbert decompositions is that they are easier to find. We test our new notion on the syzygy modules of the residue class field of $K[X_1,...,X_n]$ (as usual identified with $K$). Writing $M(n,k)$ for the $k$-th syzygy module, we show that theHilbert depth of M(n,1) is $\lfloor(n+1)/2\rfloor$. Furthermore, we show that, for $n > k \ge \lfloor n/2\rfloor$, the Hilbert depth of $M(n,k)$ is equal to $n-1$. We conjecture that the same holds for the Stanley depth. For the range $n/2 > k > 1$, it seems impossible to come up with a compact formula for the Hilbert depth. Instead, we provide very precise asymptotic results as $n$ becomes large.

math.AC↗

arXiv2010

Multivariate $p$-adic formal congruences and integrality of Taylor coefficients of mirror maps

We generalise Dwork's theory of $p$-adic formal congruences from the univariate to a multi-variate setting. We apply our results to prove integrality assertions on the Taylor coefficients of (multi-variable) mirror maps. More precisely, with $\mathbf z=(z_1,z_2,...,z_d)$, we show that the Taylor coefficients of the multi-variable series $q(\mathbf z)=z_i\exp(G(\mathbf z)/F(\mathbf z))$ are integers, where $F(\mathbf z)$ and $G(\mathbf z)+\log(z_i) F(\mathbf z)$, $i=1,2,...,d$, are specific solutions of certain GKZ systems. This result implies the integrality of the Taylor coefficients of numerous families of multi-variable mirror maps of Calabi-Yau complete intersections in weighted projective spaces, as well as of many one-variable mirror maps in the "Tables of Calabi-Yau equations" [arXiv:math/0507430] of Almkvist, van Enckevort, van Straten and Zudilin. In particular, our results prove a conjecture of Batyrev and van Straten in [Comm. Math. Phys. 168 (1995), 493-533] on the integrality of the Taylor coefficients of canonical coordinates for a large family of such coordinates in several variables.

math.NT↗

arXiv2010

Determinants of (generalised) Catalan numbers

We show that recent determinant evaluations involving Catalan numbers and generalisations thereof have most convenient explanations by combining the Lindström-Gessel-Viennot theorem on non-intersecting lattice paths with a simple determinant lemma from [Manuscripta Math. 69 (1990), 173-202]. This approach leads also naturally to extensions and generalisations.

math.CO↗

arXiv2009

The interaction of a gap with a free boundary in a two dimensional dimer system

Let $\ell$ be a fixed vertical lattice line of the unit triangular lattice in the plane, and let $\Cal H$ be the half plane to the left of $\ell$. We consider lozenge tilings of $\Cal H$ that have a triangular gap of side-length two and in which $\ell$ is a free boundary - i.e., tiles are allowed to protrude out half-way across $\ell$. We prove that the correlation function of this gap near the free boundary has asymptotics $\frac{1}{4πr}$, $r\to\infty$, where $r$ is the distance from the gap to the free boundary. This parallels the electrostatic phenomenon by which the field of an electric charge near a conductor can be obtained by the method of images.

math.CO↗

arXiv2009

Supercongruences satisfied by coefficients of 2F1 hypergeometric series

Recently, Chan, Cooper and Sica conjectured two congruences for coefficients of classical 2F1 hypergeometric series which also arise from power series expansions of modular forms in terms of modular functions. We prove these two congruences using combinatorial properties of the coefficients.

math.NT↗

arXiv2009

Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions

Given a finite irreducible Coxeter group $W$, a positive integer $d$, and types $T_1,T_2,...,T_d$ (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions $c=\si_1\si_2 cdots\si_d$ of a Coxeter element $c$ of $W$, such that $\si_i$ is a Coxeter element in a subgroup of type $T_i$ in $W$, $i=1,2,...,d$, and such that the factorisation is "minimal" in the sense that the sum of the ranks of the $T_i$'s, $i=1,2,...,d$, equals the rank of $W$. For the exceptional types, these decomposition numbers have been computed by the first author. The type $A_n$ decomposition numbers have been computed by Goulden and Jackson, albeit using a somewhat different language. We explain how to extract the type $B_n$ decomposition numbers from results of Bóna, Bousquet, Labelle and Leroux on map enumeration. Our formula for the type $D_n$ decomposition numbers is new. These results are then used to determine, for a fixed positive integer $l$ and fixed integers $r_1\le r_2\le ...\le r_l$, the number of multi-chains $π_1\le π_2\le ...\le π_l$ in Armstrong's generalised non-crossing partitions poset, where the poset rank of $π_i$ equals $r_i$, and where the "block structure" of $π_1$ is prescribed. We demonstrate that this result implies all known enumerative results on ordinary and generalised non-crossing partitions via appropriate summations. Surprisingly, this result on multi-chain enumeration is new even for the original non-crossing partitions of Kreweras. Moreover, the result allows one to solve the problem of rank-selected chain enumeration in the type $D_n$ generalised non-crossing partitions poset, which, in turn, leads to a proof of Armstrong's $F=M$ Conjecture in type $D_n$.

math.CO↗

arXiv2009

On the integrality of the Taylor coefficients of mirror maps, II

We continue our study begun in "On the integrality of the Taylor coefficients of mirror maps" (arXiv:0907.2577) of the fine integrality properties of the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z) {\bf F}(z)$ are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at $z=0$. More precisely, we address the question of finding the largest integer $v$ such that the Taylor coefficients of $(z ^{-1}{\bf q}(z))^{1/v}$ are still integers. In particular, we determine the Dwork-Kontsevich sequence $(u_N)_{N\ge1}$, where $u_N$ is the largest integer such that $q(z)^{1/u_N}$ is a series with integer coefficients, where $q(z)=\exp(G(z)/F(z))$, $F(z)=\sum_{m=0}^{\infty} (Nm)! z^m/m!^N$ and $G(z)=\sum_{m=1}^{\infty} (H_{Nm}-H_m)(Nm)! z^m/m!^N$, with $H_n$ denoting the $n$-th harmonic number, conditional on the conjecture that there are no prime number $p$ and integer $N$ such that the $p$-adic valuation of $H_N-1$ is strictly greater than 3.

math.NT↗

arXiv2009

On the integrality of the Taylor coefficients of mirror maps

math.NT↗

arXiv2009

On the integrality of the Taylor coefficients of mirror maps

We show that the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$ are integers, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z) {\bf F}(z)$ are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at $z=0$. We also address the question of finding the largest integer $u$ such that the Taylor coefficients of $(z ^{-1}{\bf q}(z))^{1/u}$ are still integers. As consequences, we are able to prove numerous integrality results for the Taylor coefficients of mirror maps of Calabi-Yau complete intersections in weighted projective spaces, which improve and refine previous results by Lian and Yau, and by Zudilin. In particular, we prove the general ``integrality'' conjecture of Zudilin about these mirror maps. A further outcome of the present study is the determination of the Dwork-Kontsevich sequence $(u_N)_{N\ge1}$, where $u_N$ is the largest integer such that $q(z)^{1/u_N}$ is a series with integer coefficients, where $q(z)=\exp(F(z)/G(z))$, $F(z)=\sum_{m=0} ^{\infty} (Nm)! z^m/m!^N$ and $G(z)=\sum_{m=1} ^{\infty} (H_{Nm}-H_m)(Nm)! z^m/m!^N$, with $H_n$ denoting the $n$-th harmonic number, conditional on the conjecture that there are no prime number $p$ and integer $N$ such that the $p$-adic valuation of $H_N-1$ is strictly greater than 3.

math.NT↗

arXiv2009

Démonstration de l'Observation 2 d'Almkvist et Zudilin

We prove Obnservation 2 in arXiv:math/0402386 by Gert Almkvist and Wadim Zudilin.

math.NT↗

arXiv2009

Euler characteristic of the truncated order complex of generalized noncrossing partitions

The purpose of this note is to complete the study, begun in the first author's PhD thesis, of the topology of the poset of generalized noncrossing partitions associated to real reflection groups. In particular, we calculate the Euler characteristic of this poset with the maximal and minimal elements deleted. As we show, the result on the Euler characteristic extends to generalized noncrossing partitions associated to well-generated complex reflection groups.

math.CO↗

arXiv2008

On the non-quadraticity of values of the q-exponential function and related q-series

We investigate arithmetic properties of values of the entire function $$ F(z)=F_q(z;λ)=\sum_{n=0}^\infty\frac{z^n}{\prod_{j=1}^n(q^j-λ)}, \qquad |q|>1, \quad λ\notin q^{\mathbb Z_{>0}}, $$ that includes as special cases the Tschakaloff function ($λ=0$) and the $q$-exponential function ($λ=1$). In particular, we prove the non-quadraticity of the numbers $F_q(α;λ)$ for integral $q$, rational $λ$ and $α\notin-λq^{\mathbb Z_{>0}}$, $α\ne0$.

math.NT↗

arXiv2008

A factorization theorem for classical group characters, with applications to plane partitions and rhombus tilings

We prove that a Schur function of rectangular shape $(M^n)$ whose variables are specialized to $x_1,x_1^{-1},...,x_n,x_n^{-1}$ factorizes into a product of two odd orthogonal characters of rectangular shape, one of which is evaluated at $-x_1,...,-x_n$, if $M$ is even, while it factorizes into a product of a symplectic character and an even orthogonal character, both of rectangular shape, if $M$ is odd. It is furthermore shown that the first factorization implies a factorization theorem for rhombus tilings of a hexagon, which has an equivalent formulation in terms of plane partitions. A similar factorization theorem is proven for the sum of two Schur functions of respective rectangular shapes $(M^n)$ and $(M^{n-1})$.

math.CO↗

arXiv2008

On a linear form for Catalan's constant

It is shown how Andrews' multidimensional extension of Watson's transformation between a very-well-poised $_8ϕ_7$-series and a balanced $_4ϕ_3$-series can be used to give a straightforward proof of a conjecture of Zudilin and the second author on the arithmetic behaviour of the coefficients of certain linear forms of 1 and Catalan's constant. This proof is considerably simpler and more stream-lined than the first proof, due to the second author.

math.NT↗

arXiv2008

Parity patterns associated with lifts of Hecke groups

Let $q$ be an odd prime, $m$ a positive integer, and let $\Ga_m(q)$ be the group generated by two elements $x$ and $y$ subject to the relations $x^{2m}=y^{qm}=1$ and $x^2=y^q$; that is, $\Ga_m(q)$ is the free product of two cyclic groups of orders $2m$ respectively $qm$, amalgamated along their subgroups of order $m$. Our main result determines the parity behaviour of the generalized subgroup numbers of $\Ga_m(q)$ which were defined in [T. W. Müller, Adv. in Math. 153 (2000), 118-154], and which count all the homomorphisms of index $n$ subgroups of $\Ga_m(q)$ into a given finite group $H$, in the case when $\gcd(m,| H|)=1$. This computation depends upon the solution of three counting problems in the Hecke group $\mathfrak H(q)=C_2*C_q$: (i) determination of the parity of the subgroup numbers of $\mathfrak H(q)$; (ii) determination of the parity of the number of index $n$ subgroups of $\mathfrak H(q)$ which are isomorphic to a free product of copies of $C_2$ and of $C_\infty$; (iii) determination of the parity of the number of index $n$ subgroups in $\mathfrak H(q)$ which are isomorphic to a free product of copies of $C_q$. The first problem has already been solved in [T. W. Müller, in: {\it Groups: Topological, Combinatorial and Arithmetic Aspects}, (T. W. Müller ed.), LMS Lecture Notes Series 311, Cambridge University Press, Cambridge, 2004, pp. 327-374]. The bulk of our paper deals with the solution of Problems (ii) and (iii).

math.GR↗

arXiv2008

Asymptotics for random walks in alcoves of affine Weyl groups

Asymptotic results are derived for the number of random walks in alcoves of affine Weyl groups (which are certain regions in $n$-dimensional Euclidean space bounded by hyperplanes), thus solving problems posed by Grabiner [J. Combin. Theory Ser. A 97 (2002), 285-306]. These results include asymptotic expressions for the number of vicious walkers on a circle, and as well for the number of vicious walkers in an interval. The proofs depart from the exact results of Grabiner [loc. cit.], and require as diverse means as results from symmetric function theory and the saddle point method, among others.

math.CO↗

arXiv2006

The $M$-triangle of generalised non-crossing partitions for the types $E_7$ and $E_8$

The $M$-triangle of a ranked locally finite poset $P$ is the generating function $\sum_{u,w\in P} ^{}μ(u,w) x^{\rk u}y^{\rk w}$, where $μ(.,.)$ is the Möbius function of $P$. We compute the $M$-triangle of Armstrong's poset of $m$-divisible non-crossing partitions for the root systems of type $E_7$ and $E_8$. For the other types except $D_n$ this had been accomplished in the earlier paper "The $F$-triangle of the generalised cluster complex." Altogether, this almost settles Armstrong's $F=M$ Conjecture predicting a surprising relation between the $M$-triangle of the $m$-divisible partitions poset and the $F$-triangle (a certain refined face count) of the generalised cluster complex of Fomin and Reading, the only gap remaining in type $D_n$. Moreover, we prove a reciprocity result for this $M$-triangle, again with the possible exception of type $D_n$. Our results are based on the calculation of certain decomposition numbers for the reflection groups of types $E_7$ and $E_8$, which carry in fact finer information than does the $M$-triangle. The decomposition numbers for the other exceptional reflection groups had been computed in the earlier paper. We present a conjectured formula for the type $A_n$ decomposition numbers.

math.CO↗

arXiv2006

The $F$-triangle of the generalised cluster complex

The $F$-triangle is a refined face count for the generalised cluster complex of Fomin and Reading. We compute the $F$-triangle explicitly for all irreducible finite root systems. Furthermore, we use these results to partially prove the "$M=F$ Conjecture" of Armstrong which predicts a surprising relation between the $F$-triangle and the Möbius function of his $m$-divisible partition poset associated to a finite root system.

math.CO↗