Narrow Results

We propose a new generalization of the $r$-Stirling numbers of the first kind and their analogs. These numbers appear as specialization of a new class of symmetric function, and they can be seen as a natural generalizations of the $r$-Stirling numbers of the first kind and their analogs. We also give a combinatorial interpretations of the classical numbers in terms of $s$-tuples of permutations of $[n]$ with $k$ cycles where the first $r$ elements of each permutation are in distinct cycles, and using the inversion statistics on the cycles in the cases of the analogs numbers. Moreover, using the hyperharmonic numbers and the $r$-Stirling numbers of the first kind new formulas and useful properties are established.

We show the use of a reconfigurable computer in computing the correlation immunity of Boolean functions of up to 6 variables. Boolean functions with high correlation immunity are desired in cryptographic systems because they are immune to correlation attacks. The SRC-6 reconfigurable computer was programmed in Verilog to compute the correlation immunity of functions. This computation is performed at a rate that is 190 times faster than a conventional computer. Our analysis of the correlation immunity is across all n-variable Boolean functions, for 2 ≤ n ≤ 6, thus obtaining, for the first time, a complete distribution of such functions. We also compare correlation immunity with two other cryptographic properties, nonlinearity and degree.

In this paper, we construct the quantum fields of three types of universal characters corresponding to partition shapes $\pi =(3)$, $\pi =(2,1)$ and $\pi =(1^3)$ and derive the commutation relations of quantum fields. By acting the quantum fields on the constant function $1$, the generating functions of three types of universal characters have been investigated. Furthermore, we discuss three integrable systems characterized by $(3)$-type, $(2,1)$-type and $(1^3)$-type universal characters.

It has been argued that Nekrasov's partition function gives the generating function of refined BPS state counting in the compactification of M theory on local Calabi–Yau spaces. We show that a refined version of the topological vertex we previously proposed (arXiv:hep-th/0502061) is a building block of Nekrasov's partition function with two equivariant parameters. Compared with another refined topological vertex by Iqbal, Kozcaz and Vafa (arXiv:hep-th/0701156), our refined vertex is expressed entirely in terms of the specialization of the Macdonald symmetric functions which is related to the equivariant character of the Hilbert scheme of points on ℂ². We provide diagrammatic rules for computing the partition function from the web diagrams appearing in geometric engineering of Yang–Mills theory with eight supercharges. Our refined vertex has a simple transformation law under the flop operation of the diagram, which suggests that homological invariants of the Hopf link are related to the Macdonald functions.

This article is devoted to the study of several algebras related to symmetric functions, which admit linear bases labelled by various combinatorial objects: permutations (free quasi-symmetric functions), standard Young tableaux (free symmetric functions) and packed integer matrices (matrix quasi-symmetric functions). Free quasi-symmetric functions provide a kind of noncommutative Frobenius characteristic for a certain category of modules over the 0-Hecke algebras. New examples of indecomposable H_n(0)-modules are discussed, and the homological properties of H_n(0) are computed for small n. Finally, the algebra of matrix quasi-symmetric functions is interpreted as a convolution algebra.

We find explicitly the multiplicities in the (mixed) trace cocharacter sequence of two 3 × 3 matrices over a field of characteristic 0 and show that asymptotically they behave as polynomials of seventh degree. As a consequence we obtain also the multiplicities of certain irreducible characters in the cocharacter sequence of the polynomial identities of 3 × 3 matrices.

This work lies across three areas of investigation that are by themselves of independent interest. A problem that arose in quantum computing led us to a link that tied these areas together. This link led to the calculation of some Kronecker coefficients by computing constant terms and conversely the computations of certain constant terms by computing Kronecker coefficients by symmetric function methods. This led to results as well as methods for solving numerical problems in each of these separate areas.

We give an LU-decomposition of the supercharacter table of the group of n × n unipotent upper triangular matrices over 𝔽_q, into a lower-triangular matrix with entries in ℤ[q] and an upper-triangular matrix with entries in ℤ[q^-1]. To this end, we introduce a q deformation of a new power-sum basis of the Hopf algebra of symmetric functions in noncommuting variables. The decomposition is obtained from the transition matrices between the supercharacter basis, the q-power-sum basis and the superclass basis. This is similar to the decomposition of the character table of the symmetric group S_n given by the transition matrices between Schur functions, monomials and power-sums. We deduce some combinatorial results associated to this decomposition. In particular, we compute the determinant of the supercharacter table.

The Murphy operators in the Hecke algebra H_n of type A are explicit commuting elements whose sum generates the centre. They can be represented by simple tangles in the Homfly skein theory version of H_n. In this paper I present a single tangle which represents their sum, and which is obviously central. As a consequence it is possible to identify a natural basis for the Homfly skein of the annulus, .

Symmetric functions of the Murphy operators are also central in H_n. I define geometrically a homomorphism from to the centre of each algebra H_n, and find an element in , independent of n, whose image is the mth power sum of the Murphy operators. Generating function techniques are used to describe images of other elements of in terms of the Murphy operators, and to demonstrate relations among other natural skein elements.

The main goal is to find the Homfly polynomial of a link formed by decorating each component of the Hopf link with the closure of a directly oriented tangle. Such decorations are spanned in the Homfly skein of the annulus by elements Q_λ, depending on partitions λ. We show how the 2-variable Homfly invariant <λ, μ> of the Hopf link arising from decorations Q_λ and Q_μ can be found from the Schur symmetric function s_μ of an explicit power series depending on λ. We show also that the quantum invariant of the Hopf link coloured by irreducible sl(N)_q modules V_λ and V_μ, which is a 1-variable specialisation of <λ, μ>, can be expressed in terms of an N × N minor of the Vandermonde matrix (q^ij).

Many constructs in mathematical physics entail notational complexities, deriving from the manipulation of various types of index sets which often can be reduced to labelling by various multisets of integers. In this work, we develop systematically the "Dirichlet Hopf algebra of arithmetics" by dualizing the addition and multiplication maps. Then we study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, but form only a weaker Hopf gebra obeying a weakened homomorphism axiom. A careful identification of the algebraic structures involved is done featuring subtraction, division and derivations derived from coproducts and chochains using branching operators. The consequences of the weakened structure of a Hopf gebra on cohomology are explored, showing this has major impact on number theory. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf algebra is then cured by introducing an "unrenormalized" coproduct and an "unrenormalized" pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for non-coprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number theoretically meaningful Hopf gebra (literally: antipodal convolution). This can be modelled alternatively by employing Rota–Baxter operators. We stress the need for a characteristic-free development where possible, to have a sound starting point for generalizations of the algebraic structures. The last section provides three key applications: symmetric function theory, quantum (matrix) mechanics, and the combinatorics of renormalization in QFT which can be discerned as functorially inherited from the development at the number-theoretic level as outlined here. Hence the occurrence of number theoretic functions in QFT becomes natural.

We present a theory for splitting algebras of monic polynomials over rings, and apply the results to symmetric functions, and Galois theory.

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and co-free. In the process of looking for bases which generate the space we define orders on the set partitions and set compositions which allow us to define bases which have simple and natural rules for the product of basis elements.

For a monic polynomial $f$ over a commutative, unitary ring $A$ the splitting algebra $A_f$ is the universal $A$-algebra such that $f$ splits in $A_f$. The symmetric group acts on the splitting algebra by permuting the roots of $f$. It is known that if the intersection of the annihilators of the elements $2$ and $D_f$ (where $D_f$ depends on $f$) in $A$ is zero, then the invariants under the group action are exactly equal to $A$. We show that the converse holds.

For $k\le n$, let $E(2n,k)$ be the sum of all multiple zeta values with even arguments whose weight is $2n$ and whose depth is $k$. Of course $E(2n,1)$ is the value $\zeta (2n)$ of the Riemann zeta function at $2n$, and it is well known that $E(2n,2)=\frac{3}{4}\zeta (2n)$. Recently Shen and Cai gave formulas for $E(2n,3)$ and $E(2n,4)$ in terms of $\zeta (2n)$ and $\zeta (2)\zeta (2n-2)$. We give two formulas for $E(2n,k)$, both valid for arbitrary $k\le n$, one of which generalizes the Shen–Cai results; by comparing the two we obtain a Bernoulli-number identity. We also give explicit generating functions for the numbers $E(2n,k)$ and for the analogous numbers $E^{\star }(2n,k)$ defined using multiple zeta-star values of even arguments.

We show that certain averages over ensembles of truncated random unitary matrices enumerate configurations of random-turns vicious walkers on ℤ.

Various infinite-dimensional versions of Calogero-Moser operator are discussed in relation with the theory of symmetric functions and representation theory of basic classical Lie superalgebras.