Twisted Vertex Operators And Shifted Schur Measures

On the measure space (D, F, PS S), for each partition (1,2)(12), wedefine the probability measure PS S=Q (x) P (x)/Z, where Q (x)and P (x)is the SchurQ-functions, Z is a normalization constant, and D is the set of all strict partitions：D: Dn0Dn（Dn is the set of all strict partitions of n）. We call this measure the shiftedSchur measure, which is analogous to the much-studied Schur measure. For Schur measure,Johansson had achieved a valuable research result: a limit law for1h, which is basedon specialization that the first m coordinates of x and the first n coordinates of y equalto and the rest coordinates of both x and y equal to zero. In light of the limit law for Schurmeasures, Tracy has derived the limit law for the shifted Schur measure in2008, which isanalogous to the limit law of Schur measures. In this paper, for the shifted Schur measure,we study the limit law by the tool of twisted vertex operators.Vertex operators have a very wide range of applications: N Jing has used vertex operatormethod to research the Schur function, Schur Q-function, the Hall-Littlewood function in1991, which enlighten us to research the shifted Schur measures by approach of vertexoperators. Tracy has derived the limit law for shifted Schur measure in2008, and in thispaper, we make the vertex operators realization of Schur Q–function by constructing propervertex operators, by which we have improved the method of Tracy. In the process, we presentthe Pfaffian realization of the correlation functions of the shifted Schur measure with the toolof vertex operators, and then offer the proof of the limit laws by vertex operators.The main ideas of this article are as follows:(1) Firstly, this paper gives the concepts and properties of the partitions, the vertex operators,Schur Q–functions, the vertex operators realization of it and so on, which makes someessential prepares for the research in follows;(2) Secondly, this paper presents the results of the limits law of the Schur measure and theshifted Schur measure. On the basis of Tracy’s work, we have improved the process ofproving the limit for the shifted Schur measure by vertex operators; (3) Finally, the main contents of the article are: the applications of the vertex operators, andthe supplementary applications of the Pfaffian, then present the Pfaffian realization of thecorrelation functions of the shifted Schur measure with the tool of vertex operators, whichmake some fundamental prepares for the proof of the limits law for the shifted Schurmeasure.