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In this paper, let F q be a finite field with q = 2 n elements and let [ A ] ∈ P G L 2 (F q) be of order 2 or 3, where A = a b 1 d . We determine all invariant irreducible (monic) polynomials by the action of [ A ] ∈ P G L 2 (F q) and have irreducible factorizations of polynomials F s (x) = x q s + 1 + d x q s + a x + b over F q in two cases: (1) s = t e and t is an odd prime; (2) s = t 1 t 2 and t 1 , t 2 are two distinct odd primes. Moreover, we construct some binary irreducible quasi-cyclic expurgated Goppa codes and extended Goppa codes by invariant irreducible polynomials under the action of [ A ] ∈ P G L 2 (F q) . [ABSTRACT FROM AUTHOR]

Abstract Random matrix products arise in many science and engineering problems. An efficient evaluation of its growth rate is of great interest to researchers in diverse fields. In the current paper, we reformulate this problem with a generating function approach, based on which two analytic methods are proposed to compute the growth rate. The new formalism is demonstrated in a series of examples including an Ising model subject to on-site random magnetic fields, which seems very efficient and easy to implement. Through an extensive comparison with numerical computation, we see that the analytic results are valid in a region of considerable size.The formulation could be conveniently applied to stochastic processes with more complex structures.

Random matrix products arise in many science and engineering problems. An efficient evaluation of its growth rate is of great interest to researchers in diverse fields. In the current paper, we reformulate this problem with a generating function approach, based on which two analytic methods are proposed to compute the growth rate. The new formalism is demonstrated in a series of examples including an Ising model subject to on-site random magnetic fields, which seems very efficient and easy to implement. Through an extensive comparison with numerical computation, we see that the analytic results are valid in a region of considerable size.The formulation could be conveniently applied to stochastic processes with more complex structures. [ABSTRACT FROM AUTHOR]

The irreducible representations of symmetric groups can be realized as certain graded pieces of invariant rings, equivalently as global sections of line bundles on partial flag varieties. There are various ways to choose useful bases of such Specht modules S λ. Particularly powerful are web bases, which make important connections with cluster algebras and quantum link invariants. Unfortunately, web bases are only known in very special cases—essentially, only the cases λ = (d , d) and λ = (d , d , d). Building on work of B. Rhoades (2017), we construct an apparent web basis of invariant polynomials for the 2-parameter family of Specht modules with λ of the form (d , d , 1 ℓ). The planar diagrams that appear are noncrossing set partitions, and we thereby obtain geometric interpretations of earlier enumerative results in combinatorial dynamics. [ABSTRACT FROM AUTHOR]