Odd Rank Research Articles

The Weil bound for generalized Kloosterman sums of half-integral weight

Abstract Let L be an even lattice of odd rank with discriminant group L ′ / L {L^{\prime}/L} , and let α , β ∈ L ′ / L {\alpha,\beta\in L^{\prime}/L} . We prove the Weil bound for the Kloosterman sums S α , β ⁢ ( m , n , c ) {S_{\alpha,\beta}(m,n,c)} of half-integral weight for the Weil Representation attached to L. We obtain this bound by proving an identity that relates a divisor sum of Kloosterman sums to a sparse exponential sum. This identity generalizes Kohnen’s identity for plus space Kloosterman sums with the theta multiplier system.

On [formula omitted]-neighborly reorientations of oriented matroids

We study the existence and the number of k-neighborly reorientations of an oriented matroid. This leads to k-variants of McMullen’s problem and Roudneff’s conjecture, the case k=1 being the original statements. Adding to results of Larman and García-Colín, we provide new bounds on k-McMullen’s problem and prove the conjecture for several ranks and k by computer. Further, we show that k-Roudneff’s conjecture for fixed rank and k reduces to a finite case analysis. As a consequence we prove the conjecture for odd rank r and k=r−12 as well as for rank 6 and k=2 with the aid of the computer.

Hamiltonian Intervals in the Lattice of Binary Paths

Let $\mathcal{P}_n$ be the set of all binary paths (i.e., lattice paths with upsteps $u = (1,1)$ and downsteps $d = (1,-1)$) of length $n$ endowed with the pointwise partial ordering (i.e., $P \le Q$ iff the lattice path $P$ lies weakly below $Q$) and let $G_n$ be its Hasse graph. For each path $P \in \mathcal{P}_n$, we denote by $I(P)$ the interval which contains the elements of $\mathcal{P}_n$ less than or equal to $P$, excluding the first two elements of $\mathcal{P}_n$, and by $G(P)$ the subgraph of $G_n$ induced by $I(P)$. In this paper, it is shown that $G(P)$ is Hamiltonian iff $P$ contains at least two peaks and $I(P)$ has equal number of elements with even and odd rank. The last condition is always true for paths ending with an upstep, whereas, for paths ending with a downstep, a simple characterization is given, based on the structure of the path.

Asymptotics for symmetrized positive moments of odd ranks

Abstract In 2007, Andrews introduced Durfee symbols and k-marked Durfee symbols so as to give a combinatorial interpretation for the symmetrized moment function $\eta _{2k}(n)$ of ranks of partitions. He also considered the relations between odd Durfee symbols and the mock theta function $\omega (q)$ , and proved that the $2k$ th moment function $\eta _{2k}^0(n)$ of odd ranks of odd Durfee symbols counts $(k+1)$ -marked odd Durfee symbols of n. In this paper, we first introduce the definition of symmetrized positive odd rank moments $\eta _k^{0+}(n)$ and prove that for all $1\leq i\leq k+1$ , $\eta _{2k-1}^{0+}(n)$ is equal to the number of $(k+1)$ -marked odd Durfee symbols of n with the ith odd rank equal to zero and $\eta _{2k}^{0+}(n)$ is equal to the number of $(k+1)$ -marked Durfee symbols of n with the ith odd rank being positive. Then we calculate the generating functions of $\eta _{k}^{0+}(n)$ and study its asymptotic behavior. Finally, we use Wright’s variant of the Hardy–Ramanujan circle method to obtain an asymptotic formula for $\eta _{k}^{0+}(n)$ .

ODD RANK VECTOR BUNDLES IN ETA-PERIODIC MOTIVIC HOMOTOPY THEORY

AbstractWe observe that, in the eta-periodic motivic stable homotopy category, odd rank vector bundles behave to some extent as if they had a nowhere vanishing section. We discuss some consequences concerning$\operatorname {\mathrm {SL}}^c$-orientations of motivic ring spectra and the étale classifying spaces of certain algebraic groups. In particular, we compute the classifying spaces of diagonalisable groups in the eta-periodic motivic stable homotopy category.

Hook length and symplectic content in partitions

The dimension of an irreducible representation of GL(n,C), Sp(2n), or SO(n) is given by the respective hook length and content formulas for the corresponding partition. The first author, inspired by the Nekrasov-Okounkov formula, conjectured combinatorial interpretations of analogous expressions involving hook lengths and symplectic/orthogonal contents. We prove special cases of these conjectures. In the process, we show that partitions of n with all symplectic contents non-zero are equinumerous with partitions of n into distinct even parts. We also present Beck-type companions to this identity. In this context, we give the parity of the number of partitions into distinct parts with odd (respectively, even) rank. We study the connection between the sum of hook lengths and the sum of inversions in the binary representation of a partition. In addition, we introduce a new partition statistic, the x-ray list of a partition, and explore its connection with distinct partitions as well as partitions maximally contained in a given staircase partition.

Ulrich bundles on Del Pezzo threefolds

We prove that for any r⩾2 the moduli space of stable Ulrich bundles of rank r and determinant OX(r) on any smooth Fano threefold X of index two is smooth of dimension r2+1 and that the same holds true for even r when the index is four, in which case no odd–rank Ulrich bundles exist. In particular this shows that any such threefold is Ulrich wild. As a preliminary result, we give necessary and sufficient conditions for the existence of Ulrich bundles on any smooth projective threefold in terms of the existence of a curve in the threefold enjoying special properties.

The number of P-vertices for acyclic matrices with given nullity

In this paper, we completely characterize those trees on n vertices for which there is a singular matrix with nullity k and the number of P-vertices is n−k−1 or n−k−2. The characterization of acyclic matrices, with rank r and the number of P-vertices is r−1, or with odd rank r and the number of P-vertices is r−2, was first investigated in Fonseca et al. (2021) [10]. Here we introduce a unified method to revisit those results, and further cover the unknown case with even rank r and the number of P-vertices being r−2. In the end, a continuity problem is fully solved.

Notes on Ramond-Ramond spinors and bispinors in double field theory

The Ramond-Ramond sector of double field theory (DFT) can be described either as an O(D, D) spinor or an O(D − 1, 1) × O(1, D − 1) bispinor. Both formulations may be related to the standard polyform expansion in terms of even or odd rank field strengths corresponding to IIA or IIB duality frames. The spinor approach is natural in a (bosonic) metric formulation of DFT, while the bispinor is indispensable for supersymmetric DFT. In these notes, we show how these two approaches may be covariantly connected using a spinorial version of the DFT vielbein, which flattens an O(D, D) spinor into a bispinor. We also elaborate on details of the bispinor formulation in both even and odd D and elaborate on the distinction between the IIA/IIB/IIA*/IIB* duality frames.

On tight sets of hyperbolic quadrics

We prove that the parameter x of a tight set T of a hyperbolic quadric Q+(2n+1,q) of an odd rank n+1 satisfies x2+w(w−x)≡0modq+1, where w is the number of points of T in any generator of Q+(2n+1,q). As this modular equation should have an integer solution in w if such a T exists, this condition rules out roughly at least one half of all possible parameters x. It generalizes a previous result by the author and K. Metsch shown for tight sets of a hyperbolic quadric Q+(5,q) (also known as Cameron–Liebler line classes in PG(3,q)).

К геометрии обобщенных неголономных многообразий Кенмоцу

The concept of a generalized nonholonomic Kenmotsu manifold is introduced. In contrast to the previously defined nonholonomic Kenmotsu manifold, the manifold studied in the article is an almost normal almost contact metric manifold of odd rank. The manifold is equipped with a metric connection with torsion, which is called the canonical connection in this work. The main properties of the canonical connection are studied. The canonical connection is an analogue of the generalized Tanaka-Webster connection. In this paper, we prove that the canonical connection is the only metric connection with torsion of a special structure that pre­serves the structural 1-form and the Reeb vector field. We study the in­trinsic geometry of a generalized nonholonomic Kenmotsu manifold equipped with a canonical connection. It is proved that if a generalized nonholonomic Kenmotsu manifold is an Einstein manifold with respect to a canonical connection, then it is Ricci-flat with respect to this connec­tion. An example of a generalized nonholonomic Kenmotsu manifold that is not a nonholonomic Kenmotsu manifold is given.

Identities, inequalities and congruences for odd ranks and k-marked odd Durfee symbols

In 2007, Andrews introduced the odd rank of odd Durfee symbols. Let N0(m,n) denote the number of odd Durfee symbols of n with odd rank m, and N0(r,m;n) be the number of odd Durfee symbols of n with odd rank congruent to r modulo m. In this paper, we give the generating functions for N0(r,12;n) by utilizing some identities involving Appell-Lerch sums m(x,q,z) and a universal mock theta function g(x,q). Based on these formulas, we determine the signs of N0(r,12;4n+t)−N0(s,12;4n+t) for all 0≤r,s≤6 and 0≤t≤3. In particular, we prove that N0(2,12;4n+1)=N0(4,12;4n+1). Moreover, let Dk0(n) denote the number of k-marked odd Durfee symbols of n. Andrews conjectured that D20(8n+s) and D30(16n+t) are even with s∈{4,6} and t∈{1,9,11,13} which were confirmed by Wang. In this paper, we found new congruences for Dk0(n). In particular, for k=2 or 3, we give characterizations of n such that Dk0(n) is odd and prove that Dk0(n) take even values with probability 1 for n≥0.

Bounds for coefficients of the f(q) mock theta function and applications to partition ranks

We compute effective bounds for α(n), the Fourier coefficients of Ramanujan's mock theta function f(q) utilizing a finite algebraic formula due to Bruinier and Schwagenscheidt. We then use these bounds to prove two conjectures of Hou and Jagadeesan on the convexity and maximal multiplicative properties of the even and odd partition rank counting functions.

Inequalities for odd ranks of odd Durfee symbols

Andrews introduced odd Durfee symbols to give an interesting combinatorial interpretation of $$\omega (q)$$ invoked by MacMahon’s modular partitions, where $$\omega (q)$$ is one of the mock theta functions defined by Watson. In analogy with Dyson’s rank, Andrews defined the odd rank of an odd Durfee symbol as the number of entries in the top row minus the number of entries in the bottom row. Let $$N^0(m,n)$$ be the number of odd Durfee symbols of n with odd rank m. In this paper, we employ Wright’s circle method to give an asymptotic formula for $$N^0(m,n)$$ which implies that the inequalities $$\begin{aligned} N^0(m,n)\ge N^0(m+2,n) \end{aligned}$$ and $$\begin{aligned} N^0(m,n)\le N^0(m,n+2) \end{aligned}$$ hold for sufficient large n. Motivated by the work of Chan and Mao, we proved that the above inequalities hold for all nonnegative integers m and n.

The Cat\u2019s Cradle: deforming the higher rank E1 and {tilde{E}}_1 theories

We use 5-brane webs to study the two-dimensional space of supersymmetric mass deformations of higher rank generalizations of the 5d E1 and {tilde{E}}_1 theories. Some of the resulting IR phases are described by IR free supersymmetric gauge theories, while others correspond to interacting fixed points. The number of different phases appears to grow with the rank. The space of deformations is qualitatively different for the even and odd rank cases, but that of the even (odd) rank E1 theory is similar to that of the odd (even) rank {tilde{E}}_1 theory. One result of our analysis predicts that the supersymmetric SU(N) theory with CS level k = frac{N}{2} + 4 and a single massless antisymmetric hypermultiplet exhibits an enhanced global symmetry at the UV fixed point, given by SU(2) × SU(2) if N is even, and SU(2) × U(1) if N is odd.

On the standard conjecture for projective compactifications of Néron models of -dimensional Abelian varieties

We prove that the Grothendieck standard conjecture of Lefschetz type holds for a smooth complex projective -dimensional variety fibred by Abelian varieties (possibly, with degeneracies) over a smooth projective curve if the endomorphism ring of the generic geometric fibre is not an order of an imaginary quadratic field. This condition holds automatically in the cases when the reduction of the generic scheme fibre at some place of the curve is semistable in the sense of Grothendieck and has odd toric rank or the generic geometric fibre is not a simple Abelian variety.

On odd ranks of odd Durfee symbols

An odd Durfee symbol of [Formula: see text] is an array of positive odd integers and a subscript [Formula: see text], [Formula: see text] such that [Formula: see text], [Formula: see text], and [Formula: see text]. Andrews defined the odd rank of an odd Durfee symbol as [Formula: see text]. Let [Formula: see text] be the number of odd Durfee symbols of [Formula: see text] with odd rank congruent to [Formula: see text] modulo [Formula: see text]. We decompose the generating function of [Formula: see text] into modular and mock modular parts. Specifically, we derive some special cases of the generating functions of [Formula: see text] for [Formula: see text]. Some generating functions are related to classical mock theta functions.

Arithmetic properties of odd ranks and k-marked odd Durfee symbols

Let N0(m,n) be the number of odd Durfee symbols of n with odd rank m, and N0(a,M;n) be the number of odd Durfee symbols of n with odd rank congruent to a modulo M. We give explicit formulas for the generating functions of N0(a,M;n) and their ℓ-dissections where 0≤a≤M−1 and M,ℓ∈{2,4,8}. From these formulas, we obtain some interesting arithmetic properties of N0(a,M;n). Furthermore, let Dk0(n) denote the number of k-marked odd Durfee symbols of n. Andrews (2007) conjectured that D20(n) is even if n≡4 or 6 (mod 8) and D30(n) is even if n≡1,9,11 or 13 (mod 16). Using our results on odd ranks, we prove Andrews' conjectures.

MMP via wall-crossing for moduli spaces of stable sheaves on an Enriques surface

We use wall-crossing in the Bridgeland stability manifold to systematically study the birational geometry of the moduli space Mσ(v) of σ-semistable objects of class v for a generic stability condition σ on an arbitrary Enriques surface X. In particular, we show that for any other generic stability condition τ, the two moduli spaces Mτ(v) and Mσ(v) are birational. As a consequence, we show that for primitive v of odd rank Mσ(v) is birational to a Hilbert scheme of points. Similarly, in even rank we show that Mσ(v) is birational to a moduli space of torsion sheaves supported on a hyperelliptic curve when ℓ(v)=1. As an added bonus of our work, we prove that the Donaldson-Mukai map θv,σ:v⊥→Pic(Mσ(v)) is an isomorphism for these classes. Finally, we use our classification to fully describe the geometry of the only two examples of moduli of stable sheaves on X that are uniruled (and thus not K-trivial).

Birational geometry of moduli spaces of stable objects on Enriques surfaces

Using wall-crossing for K3 surfaces, we establish birational equivalence of moduli spaces of stable objects on generic Enriques surfaces for different stability conditions. As an application, we prove in the case of a Mukai vector of odd rank that they are birational to Hilbert schemes. The argument makes use of a new Chow-theoretic result, showing that moduli spaces on an Enriques surface give rise to constant cycle subvarieties of the moduli spaces of the covering K3.