On flushed partitions and concave compositions

In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution with modifications, we prove several problems of different nature, including Andrews' partition identities involving initial repetitions and partition theoretical interpretations of three mock theta functions of third order $f(q)$, $\phi(q)$ and $\psi(q)$. An identity of Ramanujan is proved combinatorially. Several new identities are also established.Comment: 19 page

Parameters for Twisted Representations

The study of Hermitian forms on a real reductive group $G$ gives rise, in the unequal rank case, to a new class of Kazhdan-Lusztig-Vogan polynomials. These are associated with an outer automorphism $\delta$ of $G$, and are related to representations of the extended group $$. These polynomials were defined geometrically by Lusztig and Vogan in "Quasisplit Hecke Algebras and Symmetric Spaces", Duke Math. J. 163 (2014), 983--1034. In order to use their results to compute the polynomials, one needs to describe explicitly the extension of representations to the extended group. This paper analyzes these extensions, and thereby gives a complete algorithm for computing the polynomials. This algorithm is being implemented in the Atlas of Lie Groups and Representations software

On The Structure Of The Chan-Paton Factors For D-Branes In Type II Orientifolds

We determine the structure of the Chan-Paton factors of the open strings ending on space filling D-branes in Type II orientifolds. Through the analysis, we obtain a rule concerning possible distribution of O-plane types. The result is applied to classify the topology of D-branes in terms of Fredholm operators and K-theory, deriving a proposal made earlier and extending it to more general cases. It is also applied to compactifications with N=1 supersymmetry in four-dimensions. We adapt and develop the language of category in this context, and use it to describe some decay channels.Comment: 137 page