Littlewood-Richardson rules for ordinary and projective representations of symmetric group

The main result of this thesis is a new shifted Littlewood-Richardson rule. In 1934 Littlewood and Richardson stated a combinatorial interpretation for the coefficients arising from the decomposition of the induction product of representations of symmetric groups. Much later D. White proved a theorem leading to an alternative description of these coefficients. These rules provide examples of the rich relationships between the representation theory of symmetric groups, symmetric functions, and tableau combinatorics. The theory of projective representations of symmetric groups also has strong connections with symmetric functions and the shifted tableau theory of Sagan and Worley. Recently Stembridge gave the first analogue of the Littlewood-Richardson rule in the context of projective representations of symmetric groups. This rule computes the decomposition of an induced product of projective representations of symmetric groups, where the multiplicities are expressed in terms of shifted tableaux. This thesis presents a new version of Stembridge's shifted rule. This new shifted rule allows for the definition of an involution on generalized shifted tableaux which shows directly that the definition of the skew Schur Q functions as the generating function of generalized shifted tableaux, yields a symmetric function. This involution is a shifted analogue of the "automorphisms of conjugation" given by Lascoux and Schutzenberger. As in many of the proofs of the classical rule, the main tools employed are the Robinson-Schensted-Knuth correspondence and Schutzenberger's jue de taquin, together with their shifted analogues. Much of this theory is revisited, with particular emphasis on the connection of slides with recording tableaux of shuffles of Schensted row and column insertions. The last chapter gives a unified proof of the classical rule, its aforementioned alternative version, and their extensions. This proof uses a single directly defined involution. In the context of these extensions, the Littlewood-Richardson rule and the alternative rule are seen to be identical. This symmetry does not seem to extend to the shifted case.