Irreducibility criterion for a finite-dimensional highest weight representation of the sl(2) loop algebra and the dimensions of reducible representations

We present a necessary and sufficient condition for a finite-dimensional highest weight representation of the $sl_2$ loop algebra to be irreducible. In particular, for a highest weight representation with degenerate parameters of the highest weight, we can explicitly determine whether it is irreducible or not. We also present an algorithm for constructing finite-dimensional highest weight representations with a given highest weight. We give a conjecture that all the highest weight representations with the same highest weight can be constructed by the algorithm. For some examples we show the conjecture explicitly. The result should be useful in analyzing the spectra of integrable lattice models related to roots of unity representations of quantum groups, in particular, the spectral degeneracy of the XXZ spin chain at roots of unity associated with the $sl_2$ loop algebra.Comment: 32 pages with no figure; with corrections on the published versio

Writing representations over proper division subrings

Let �� be a division ring, and G a finite group of automorphisms of E whose elements are distinct modulo inner automorphisms of ��. Let �� = ��G be the division subring of elements of �� fixed by G. Given a representation p : �� →��d×d of an �� -algebra ��, we give necessary and sufficient conditions for p to be writable over ��. (Here ��d×d denotes the algebra of d×d matrices over ��, and a matrix A writes p over �� if A−1p(��)A ⊆ Fd×d.) We give an algorithm for constructing an A, or proving that no A exists. The case of particular interest to us is when �� is a field, and p is absolutely irreducible. The algorithm relies on an explicit formula for A, and a generalization of Hilbert’s Theorem 90 that arises in galois cohomology. The algorithm has applications to the construction of absolutely irreducible group representations (especially for solvable groups), and to the recognition of class C5 in Aschbacher’s matrix group classification scheme [1, 13]

Constructing irreducible representations of finitely presented algebras

By combining well-known techniques from both noncommutative algebra and computational commutative algebra, we observe that an algorithmic approach can be applied to the study of irreducible representations of finitely presented algebras. In slightly more detail: Assume that $n$ is a positive integer, that $k$ is a computable field, that $\bar{k}$ denotes the algebraic closure of $k$, and that $M_n(\bar{k})$ denotes the algebra of $n \times n$ matrices with entries in $\bar{k}$. Let $R$ be a finitely presented $k$-algebra. Calculating over $k$, the procedure will (a) decide whether an irreducible representation $R \to M_n(\bar{k})$ exists, and (b) explicitly construct an irreducible representation $R \to M_n(\bar{k})$ if at least one exists. (For (b), it is necessary to assume that $k[x]$ is equipped with a factoring algorithm.) An elementary example is worked through.Comment: 9 pages. Final version. To appear in J. Symbolic Computatio

Fast Fourier Transforms for Finite Inverse Semigroups

We extend the theory of fast Fourier transforms on finite groups to finite inverse semigroups. We use a general method for constructing the irreducible representations of a finite inverse semigroup to reduce the problem of computing its Fourier transform to the problems of computing Fourier transforms on its maximal subgroups and a fast zeta transform on its poset structure. We then exhibit explicit fast algorithms for particular inverse semigroups of interest--specifically, for the rook monoid and its wreath products by arbitrary finite groups.Comment: ver 3: Added improved upper and lower bounds for the memory required by the fast zeta transform on the rook monoid. ver 2: Corrected typos and (naive) bounds on memory requirements. 30 pages, 0 figure