We cover some of the foundational results of representation theory including Maschke’s Theorem, Schur’s Lemma, and the Schur Orthogonality Relations. We consider character theory, constructions of representations, and conjugacy classes. Finally, we touch upon Pontryagin Duality for abelian groups and Young Tableaux for the Symmetric Group.

The classical Wiener’s lemma and its various generalizations are important and have numerous applications in numerical analysis, wavelet theory, frame theory, and sampling theory. There are many different equivalent formulations for the classical Wiener’s lemma, with an equivalent formulation suitable for our generalization involving commutative algebra of infinite matrices W := {(a(j − j))j,j′...

Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/repns/notes 2014-15/03 generalities finite.pdf] 1. Subrepresentations, complete reducibility, unitarization 2. Dual/contragredient representations 3. Regular and biregular representations L(G) 4. Schur’s lemma 5. Central characters of irreducibles 6. Tensor products of repres...

The Springer modules have a combinatorial property called " coincidence of dimensions, " i.e., the Springer modules are naturally decomposed into submodules with common dimensions. Morita and Nakajima proved the property by giving modules with common dimensions whose induced modules are isomorphic to the submodules of Springer modules. They proved that the induced modules are isomorphic to the ...

We obtain a tableau definition of the skew Schubert polynomials named by Lascoux, which are defined as flagged double skew Schur functions. These polynomials are in fact Schubert polynomials in two sets of variables indexed by 321-avoiding permutations. From the divided difference definition of the skew Schubert polynomials, we construct a lattice path interpretation based on the Chen-Li-Louck ...

Domain decomposition of the network conductance matrix is one efficient approaches to solve large-scale networks in parallel, wherein most commonly-used non-iterative method Schur complement (SC) method. However, SC could not obtain inversion directly, and computational cost will increase fast when overlapping domain expands. In this work, a novel Linking-Domain Extraction (LDE) based proposed,...

R. C. Buck fl] has shown that if a regular matrix sums every subsequence of a sequence x, then x is convergent. I. J. Maddox [4] improved Buck's theorem by showing that if a non-Schur matrix sums every subsequence of a sequence x, then x is convergent. Actually Maddox proved a stronger result: If x is divergent and A sums every subsequence of x, then A is a Schur matrix, i.e., It should be rema...