Hecke Algebra Hn Research Articles

Characters of An-1Hecke algebras at roots of unity

A Frobenius formula is presented for the characters, phi sigma p(q), of the irreducible representations, pi ( sigma ), of the An-1 Hecke algebras Hn(q) labelled by (m, k)-standard partitions, sigma , where q is a primitive pth root of unity with p=m+k. Using this result the characters phi sigma p(q) may be expressed as linear combinations of the characters, chi lambda p(q), of representations which are irreducible when q is not a root of unity. The appropriate linear combinations are found by using fusion modification rules. For 1<or=n<or=5 all the remaining characters of irreducible representations of Hn(q) are then found, allowing a complete tabulation to be made for 2<or=p<or=5, along with the corresponding decomposition matrices.

AN ALGEBRAIC APPROACH TO VERTEX MODELS AND TRANSFER MATRIX SPECTRA

We analyse the transfer matrix spectra of vertex models associated with the Lie superalgebras gl(P|M) and sl(P|M) using the representation theory of the Hecke algebra Hn(q). We develop a terminology for discussing the Bethe ansatz computation of the spectrum from this perspective. Using representations coming from these vertex models we develop some new methods for dealing with the analysis of Hecke algebras in any specialisation, including roots of unity. We also discuss the construction and spectrum of fusion models from the viewpoint of representation theory, begining a classification of the spectrum and identifying some sectors with trivial spectrum. In particular we show that the spectrum of the sl(P|M) λ-fusion model is trivial if λP+1&gt;M.

An algorithm for characters of Hecke algebras Hn(q) of type An-1

Recently the characters of irreducible representations of the Hecke algebra Hn(q) of type An-1 were identified with the transition coefficients relating q-generalized power sum symmetric functions to Schur functions (King and Wybourne 1990). Using this result, one obtains an easy combinatorial algorithm for calculating the characters.

Characters of Hecke algebras Hn(q) of type An-1

The connection between the Ocneanu trace on Hn(q) and Schur functions leads to a simple method for calculating the irreducible characters of the Hecke algebras Hn(q). The characters appear as the elements of the transition matrix relating certain generalized power sum symmetric functions to Schur functions.

Block spin transformations in the operator formulation of two-dimensional Potts models

The author shows that statistical mechanical models, such as the q-state Potts model, whose transfer matrix, may be written in terms of the Hecke algebra Hn(q), respond to block spin scale changes at a certain critical point in a way which may be interpreted as an algebra homomorphism. Part of the structure of the algebra, and hence part of the transfer matrix spectrum of the model, is preserved provided q=4 cos2( pi /r) (r integer). These partial realisations of scale invariance are of a distinct type for each r, characterised in some cases by an endomorphism of the algebra.