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We investigate the weighted fusion category algebra *(<i>b</i>) of a block <i>b</i> of a finite group, which is defined by Markus Linckelmann based on the fusion system of the block <i>b</i> to reformulate Alperin’s weight conjecture.  We present the definition and fundamental properties of the weighted fusion category algebras from the first principle.  In particular, we give an alternative proof that they are quasi-hereditary, and show that they are Morita equivalent to their Ringel duals.  We compute the structure of the weighted fusion category algebras of tame blocks and principal 2-blocks of GL<i>_n</i>(<i>q</i>) explicitly in terms of their quivers with relations and compare them with that of the <i>q</i>-Schur algebras <i>S_n</i>(<i>q</i>) for <i>q</i> odd prime powers and <i>n</i> = 2,3.  As a result, we find structural connections between them.

Hecke algebras arise naturally in the representation theory of reductive groups over finite or <i>p</i>-adic fields.  These algebras are specializations of Iwahori-Hecke algebras which can be defined in terms of a Coxeter group and a weight function without reference to reductive groups and this is the setting we are working in.  Kazhdan-Lusztig cells play a crucial role in the study of Iwahori-Hecke algebras.  The aim of this work is to study the Kazhdan-Lusztig cells in affine Weyl groups with unequal parameters.  More precisely, we show that the Kazhdan-Lusztig polynomials of an affine Weyl group are invariant under “long enough” translations, we decompose the lowest two-sided cell into left cells and we determine the decomposition of the affine Weyl group of type <i>Ğ</i>₂ into cells for a whole class of weight functions.

Let A be an augmented algebra over a semi-simple algebra S. We show that the Ext algebra of S as an A-module, enriched with its natural A-infinity structure, can be used to reconstruct the completion of A at the augmentation ideal. We use this technical result to justify a calculation in the physics literature describing algebras that are derived equivalent to certain non-compact Calabi-Yau three-folds. Since the calculation produces superpotentials for these algebras we also include some discussion of superpotential algebras and their invariants.

Let \(p\) be a rational odd prime number, \(G\) be a finite group such that \(|G|=p^am\), with \(p \nmid m\). Let \(B\) be a \(p\)-block of \(G\) with a defect group \(E\) which is an extra-special \(p\)-group of order \(p^3\) and exponent \(p\). Consider a fixed maximal \((G, B)\)-subpair \((E, b_E)\). Let \(b\) be the Brauer correspondent of \(B\) for \(N_G(E, b_E)\). For a non-negative integer \(d\), let \(k_d(B)\) denote the number of irreducible characters \(\chi\) in \(B\) which have \(\chi(1)_p=p^{a-d}\) and let \(k_d(b)\) be the corresponding number of \(b\). Various generalizations of Alperin's Weight Conjecture and McKay's Conjecture are due to Reinhard Knorr, Geoffrey R. Robinson and Everett C. Dade. We follow Geoffrey R. Robinson's approach to consider the Ordinary Weight Conjecture, and Dade's Projective Conjecture. The general question is whether it follows from either of the latter two conjectures that \(k_d(B)=k_d(b)\) for all \(d\) for the \(p\)-block \(B\). The objective of this thesis is to show that these conjectures predict that \(k_d(B)=k_d(b)\), for all non-negative integers \(d\). It is well known that \(N_G(E, b_E)/EC_G(E)\) is a \(p^'\)-subgroup of the automorphism group of \(E\). Hence, we have considered some special cases of the above question.The unique largest normal \(p\)-subgroup of \(G\), \(O_p(G)\) is the central focus of our attention. We consider the case that \(O_p(G)\) is a central \(p\)-subgroup of \(G\), as well as the case that \(O_p(G)\) is not central. In both cases, the common factor is that \(O_p(G)\) is strictly contained in the defect group of \(B\).

Picard's theorem states that a non-constant function which is meromorphic in the complex plane C omits at most two values of the extended complex plane C*. A Picard set for a family of functions F is a subset E of the plane such that every transcendental f in F takes every value of C*, with at most two exceptions, infinitely often in C-E. If f is transcendental and meromorphic in the plane, then: (i) [Hayman and others] if N is a positive integer, f^Nf' takes all finite non-zero values infinitely often; (ii) [Hayman] either f takes every finite value infinitely often, or each derivative f^(k) takes every finite non-zero value infinitely often. We can seek analogues of Picard sets ie subsets E of the plane and an associated family of functions F, such that (for case (i)) f^Nf' takes all finite non-zero values infinitely often in C-E, for all f in F. Similarly for case (ii). In this thesis we improve or extend the results previously known, both for Picard sets proper and for the analogous cases (i) and (ii) mentioned above, when the family of functions F consists of meromorphic functions which have deficient poles (in the sense of Nevanlinna).