Andrews–Gordon type identities from combinations of Virasoro characters

For p∈{3,4} and all p′>p, with p′ coprime to p, we obtain fermionic expressions for the combination χ _1,s ^p,p′+q ^Δ χ _p−1,s ^p,p′ of Virasoro (W ₂) characters for various values of s, and particular choices of Δ. Equating these expressions with known product expressions, we obtain q-series identities which are akin to the Andrews–Gordon identities. For p=3, these identities were conjectured by Bytsko. For p=4, we obtain identities whose form is a variation on that of the p=3 cases. These identities appear to be new. The case (p,p′)=(3,14) is particularly interesting because it relates not only to W ₂, but also to W ₃ characters, and offers W ₃ analogues of the original Andrews–Gordon identities. Our fermionic expressions for these characters differ from those of Andrews et al. which involve Gaussian polynomials. BF is partially supported by grant number RFBR 05-01-01007, and OF by the Australian Research Council (ARC).     

A probabilistic proof of the Andrews-Gordon identities

Recently, Fulman proved the “extreme” cases of the Andrews-Gordon identities using a Markov chain on the non-negative integers. Here we extend Fulman's methods to prove the Andrews-Gordon identities in full generality.     

Infinite independent systems of the identities of the associative algebra over an infinite field of characteristic <Emphasis Type="Italic">p</Emphasis> > 0

In this paper some infinitely based varieties of groups are constructed and these results are transferred to the associative algebras (or Lie algebras) over an infinite field of an arbitrary positive characteristic.     

-matrices satisfy Newton's inequalities

Newton's inequalities are shown to hold for the normalized coefficients of the characteristic polynomial of any - or inverse -matrix. They are derived by establishing first an auxiliary set of inequalities also valid for both of these classes. They are also used to derive some new necessary conditions on the eigenvalues of nonnegative matrices.

     

New Identities for Hall-Littlewood Polynomials and Applications

Starting from Macdonald's summation formula of Hall-Littlewood polynomials over bounded partitions and its even partition analogue, Stembridge (Trans. Amer. Math. Soc., 319(2), (1990) 469–498) derived sixteen multiple q-identities of Rogers–Ramanujan type. Inspired by our recent results on Schur functions (Adv. Appl. Math., 27, (2001) 493–509) and based on computer experiments we obtain two further such summation formulae of Hall-Littlewood polynomials over bounded partitions and derive six new multiple q-identities of Rogers–Ramanujan type. 2000 Mathematics Subject Classification: Primary–05A19; Secondary–05A17, 05A30     

All Varieties of Normal Bands with Involution

In the present paper we completely determine the lattice L(NB ^*) of all varieties of normal bands equipped with an involutorial antiautomorphism as a fundamental operation.     

A Sequence of Unique Quaternary Griesmer Codes

This paper establishes that there is no [98,5,72]₄ code. Such a code would meet the Griesmer bound and the weights of its codewords would all be divisible by 4. The proof of nonexistence uses the uniqueness of codes with parameters [n,4,n - 5]₄,14 n 17. The uniqueness of these codes for n 15 had been established geometrically by others; but it is rederived here, along with that of the [14,4,9]₄ code, by exploiting the Hermitian form obtained when the weight function is read modulo 2.     

The Einstein 3-form Gα and its Equivalent 1-form Lα in Riemann–Cartan Space

We motivate the definition of the Einstein 3-form G by means of the contracted 2nd Bianchi identity. This definition contains the whole curvature 2-form. The L 1-form, defined via G = L ^*( ) ( is the Hodge-star, the coframe), is equivalent to the Einstein 3-form and contains all the information of the curvature 2-form relevant for the definition of the Einstein 3-form. A variational formula of Salgado on quadratic invariants of the L 1-form is discussed, generalized, and put into proper perspective.     

Variants of the Andrews-Gordon Identities

The object of this paper is to propose and prove a new generalization of the Andrews-Gordon Identities, extending a recent result of Garrett, Ismail and Stanton. We also give a combinatorial discussion of the finite form of their result which appeared in the work of Andrews, Knopfmacher, and Paule.     

A Ring Variety without an Independent Basis

An example of a ring variety without an independent basis is constructed. It is proved that this variety is the intersection of two independently based varieties.