The Schur Q-functions And Disjoint Combination Of Nature, Walking

The thesis mainly studies some combinatorial properties of Schur's Q-functions and vicious walks.The srank theory on Schur's Q-functions,parallel to the theory of rank and zrank for Schur functions,was recently introduced and developed by Clifford.It is of central interest in algebraic combinatorics,as well as in projective representation theory.Vicious walk or nonintersecting path is another important object in combinatorics.The vicious walker model is the system of one-dimensional symmetric simple random walks,in which none of walkers has met others in a given time period.It was first introduced to the mathematical physics literature by Fisher in 1984.In recent years,it has been extensively studied by statistical mechanics community as well as the combinatorics community.The thesis has two parts.The first part is to investigate the srank problem on Schur's Q-functions,where some combinatorial properties of srank are presented. The second part is to study the enumeration problem of vicious walkers,in which combinatorial derivations of the formulas for the number of configurations of three and four vicious walkers are obtained.Chapter 2 and Chapter 3 constitute the first part of this thesis.In Chapter 2, we give an overview of Schur functions,including the algebraic and combinatorial definitions of Schur functions,the connection of Schur functions and irreducible representations of symmetric groups,the Murnaghan-Nakayama rule,and the rank and zrank theory for skew Schur functions.In Chapter 3,we investigate the srank problem on Schur's Q-functions.We show that the shifted rank,or stank,of any partitionλwith distinct parts equals the lowest degree of the terms appearing in the expansion of Schur's Q_λfunction in terms of power sum symmetric functions.This gives an affirmative answer to a conjecture of Clifford.As pointed out by Clifford,the notion of the srank can be naturally extended to a skew partitionλ/μas the minimum number of bars among the corresponding skew bar tableaux.While the srank conjecture is not valid for skew partitions,we give an algorithm to compute the srank.To conclude this chapter,we introduce the notion of szrank for a strict partitionλand propose an open problem:for any strict partitionλ,whether szrank(λ) = stank(λ) always holds?Chapter 4 is devoted to the second part of this thesis.We establish a reflection principle for three lattice walks,which is used to reduce the enumeration of configurations of three vicious walkers to that of configurations of two vicious walkers.Hence we find a combinatorial interpretation of the generating function formula for the number of configurations of three vicious walkers,originally derived by Bousquet-Melou and independently by Gessel.We also find a reflection principle for four vicious walks that leads to a combinatorial interpretation of a formula due to Gessel.