By P. N. Hoffman
The learn of the symmetric teams kinds one of many uncomplicated construction blocks of recent staff concept. This booklet is the 1st thoroughly special and self-contained presentation of the wealth of knowledge referred to now at the projective representations of the symmetric and alternating teams. necessities are a simple familiarity with the ordinary thought of linear representations and a modest heritage in sleek algebra. The authors have taken pains to make sure that the entire suitable algebraic and combinatoric instruments are truly defined in this sort of method as to make the publication appropriate for graduate scholars and examine employees. After the pioneering paintings of Schur, little growth was once made for part a century on projective representations, regardless of substantial task at the similar subject of linear representations. although, within the final two decades, very important new advances have spurred extra study. This booklet develops either the early conception of Schur after which describes the most important advances that the topic has obvious due to the fact that then. particularly, the speculation of Q-functions and skew Q-functions is broadly coated that's crucial to the advance of the topic.
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Extra resources for Projective Representations of the Symmetric Groups: Q-Functions and Shifted Tableaux
26 I. INTRODUCTION (3) K , c K , implies m(K,)5 m(K,). (4) If K c , K , , then m(K)S m(K,). (5) If K, n K, = 0 , then m(K,u K,) = m(K,)+ m(K,). ur= , To establish (9,one should note that there exist positive functions in C , ( X ) vanishing on the one compact set and equal to + 1 on the other. We next define a set function p(G) on the open subsets of X by setting p(G) = sup m(K), K c G . It is not hard to check that p has generally the same properties: + 0 5 p(G) 5 rn for all G. (2) P ( 0 ) = 0.
Theorem: Let f(x) be C" on (c, d ) such that for every point x in the interval there exists an integer N , for which f'"-'(x) = 0; then f ( x ) is a polynomial. PROOF: Let G be the open set of all x for which there exists a neighborhood within which f ( x ) coincides with some polynomial, that is, a neighborhood on whichf'k'(x) vanishes identically for some value of k. Let F be the complement of G ; the theorem will be proved if we show that F is empty. The set F cannot have an isolated point, for if x, were such a point, it is the right-hand endpoint of an interval (a, x,) on which f coincides with a polynomial, and the Taylor coefficients of the polynomial coincide with the formal Taylor expansion off(x) about x,; similarly, x, is the left-hand endpoint of an interval (x,, b) on whichf(x) coincides with a polynomial, which is determined by the Taylor expansion of the function about x,; thus,fcoincides with a certain polynomial in the interval (a, b) and x, is not in F.
Theorem: Let X be a locally compact metric space and C , ( X ) the linear space of all continuous functions on X which vanish outside a compact subset of that space. If F(f)is a linear functional on C,(X) having the property that F(f)2 0 wheneverf(x) 2 0, then there exists a Radon measure p on X such that for allf PROOF: We first construct the measure p , defining the function m(K) on the class of compact subsets of X as follows: m ( K ) = inf F(u), u(x) 2 0 on X , u(x) 2 1 on K . It is fairly easy to verify that m(K) has the following properties: ( I ) 0 5 m(K) < co for all K .
Projective Representations of the Symmetric Groups: Q-Functions and Shifted Tableaux by P. N. Hoffman