Related papers: Lattice invariants from the heat kernel (II)
We give a new structural development of harmonic polynomials on Hamming space, and harmonic weight enumerators of binary linear codes, that parallels one approach to harmonic polynomials on Euclidean space and weighted theta functions of…
A natural first step in the classification of all `physical' modular invariant partition functions $\sum N_{LR}\,\c_L\,\C_R$ lies in understanding the commutant of the modular matrices $S$ and $T$. We begin this paper extending the work of…
We use the theory of vertex operator algebras and intertwining operators to obtain systems of q-difference equations satisfied by the graded dimensions of the principal subspaces of certain level k standard modules for \hat{\goth{sl}(3)}.…
In this paper, we consider the decomposition of theta series for lattice cosets of ternary lattices. We show that the natural decomposition into an Eisenstein series, a unary theta function, and a cuspidal form which is orthogonal to unary…
Let $\fg$ be any untwisted affine Kac-Moody algebra, $\mu$ any fixed complex number, and $\wt\fg(\mu)$ the corresponding toroidal extended affine Lie algebra of nullity two. For any $k$-tuple $\bm{\lambda}=({\lambda}_1, \cdots,…
We introduce and investigate an infinite family of functions which are shown to have generalised quantum modular properties. We realise their "companions" in the lower half plane both as double Eichler integrals and as non-holomorphic theta…
In this short paper, we find the transformation formula for the theta series under the action of the Jacobi modular group on the Siegel-Jacobi space. This formula generalizes the formula (5.1) obtained by Mumford in his book[p.189, Tata…
The paper contains three main results. First, we show that if a commutative semigroup variety is a modular element of the lattice Com of all commutative semigroup varieties then it is either the variety COM of all commutative semigroups or…
A series of integral lattices parametrised by integers $k,m,n$ are introduced and investigated, where $n$ is the rank of the lattice, including the root lattices described in a uniform way and unimodular lattices such as the Niemeier…
We propose a new algorithm of the finite lattice method to generate the high-temperature series for the Ising model in three dimensions. It enables us to extend the series for the free energy of the simple cubic lattice from the previous…
We construct an infinite commutative lattice of groups whose dual spaces give Kauffman finite-type invariants of long virtual knots. The lattice is based "horizontally" upon the Polyak algebra and extended "vertically" using Manturov's…
We study the minimality properties of a new type of "soft" theta functions. For a lattice $L\subset \mathbb{R}^d$, a $L$-periodic distribution of mass $\mu_L$ and an other mass $\nu_z$ centred at $z\in \mathbb{R}^d$, we define, for all…
Let $K=Q(\sqrt{-\ell})$ be an imaginary quadratic field with ring of integers $\O_K$, where $\ell$ is a square free integer such that $\ell\equiv 3 \mod 4$ and $C=[n, k]$ be a linear code defined over $\O_K/2\O_K$. The level $\ell$ theta…
We compute the critical $L$-values of some weight 3, 4, or 5 modular forms, by transforming them into integrals of the complete elliptic integral $K$. In doing so, we prove closed form formulas for some moments of $K'^3$. Many of our…
This is revised version of my preprint: Max-Plank Institut fuer Mathematik, 2001, No 16.
It was shown in previous work that the one-variable $\widehat\mu$-function defined by Zwegers (and Zagier) and his indefinite theta series attached to lattices of signature $(r\!+\!1,1)$ are both Heisenberg harmonic Maa\ss-Jacobi forms. We…
Let $F$ be a finite extension of $\mathbb{Q}_p$. Let $W(k)$ denote the Witt vectors of an algebraically closed field $k$ of characteristic $\ell$ different from $p$ and $2$, and let $\mathcal{Z}$ be the spherical Hecke algebra for $GL_n(F)$…
We characterize supersolvable lattices in terms of a certain modular type relation. McNamara and Thomas earlier characterized this class of lattices as those graded lattices having a maximal chain that consists of left-modular elements. Our…
The modular discriminant $\Delta$ is known to structure the sequence of modular forms $(M_{2k}(SL_2(\mathbb{Z})))_{k\in \; \mathbb{N}^*}$ at level $1$.\\ For all positive integer $N$, we define a strong modular unit $\Delta_N$ at level $N$…
We describe the multiplicative invariant algebras of the root lattices of all irreducible root systems under the action of the Weyl group. In each case, a finite system of fundamental invariants is determined and the class group of the…