By Emil Grosswald, Marvin Isadore Knopp, Mark Sheingorn
Emil Grosswald used to be a mathematician of serious accomplishment and noteworthy breadth of imaginative and prescient. This quantity can pay tribute to the span of his mathematical pursuits, that's mirrored within the wide selection of papers gathered the following. With contributions via best modern researchers in quantity conception, modular features, combinatorics, and similar research, this booklet will curiosity graduate scholars and experts in those fields. The prime quality of the articles and their shut connection to present learn developments make this quantity a needs to for any arithmetic library
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Extra info for A Tribute to Emil Grosswald: Number Theory and Related Analysis
Example text
These solutions were motivated by the method of Proctor [1984]. ) and this method of thinking should be an integral part of the form of the solution to the problem. 2. Answers for Sn , the Symmetric Group Most people in the field of combinatorial representation theory agree that the field begins with the fundamental results for the symmetric group Sn . We give the answers to the main questions for the case of Sn . The precise definitions of all the objects used below can be found in Appendix A2.
A) How do we index/count them? There is a bijection λ ∈ P+ 1−1 ←→ irreducible representations V λ , where P + is the cone of dominant integral weights for g (see Appendix A8). (b) What are their dimensions? The dimension of the irreducible representation V λ is dim(V λ ) = # of paths in Pπλ = α>0 λ+ρ, α , ρ, α where ρ= 1 2 α>0 α is the half sum of the positive roots, πλ is the straight line path from 0 to λ, and Pπλ = {fi1 · · · fik πλ|1 ≤ i1 , . . , ik ≤ n}, where f1 , . . , fn are the path operators introduced in [Littelmann 1995].
The unipotent varieties. Given a unipotent element u ∈ GL(n, C) with Jordan blocks given by the partition µ = (µ1 , . .