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25015 results in Abstract analysis

Page 188 of 1251

  • ON THE SYMMETRIC SQUARES OF COMPLEX AND QUATERNIONIC PROJECTIVE SPACE

  • Part of
  • YUMI BOOTE, NIGEL RAY
  • Journal:
    Glasgow Mathematical Journal / Volume 60 / Issue 3 / September 2018
    Published online by Cambridge University Press:
    13 February 2018, pp. 703-729
    Print publication:
    September 2018
    • Article
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  • The problem of computing the integral cohomology ring of the symmetric square of a topological space has long been of interest, but limited progress has been made on the general case until recently. We offer a solution for the complex and quaternionic projective spaces $\mathbb{K}$Pn, by utilising their rich geometrical structure. Our description involves generators and relations, and our methods entail ideas from the literature of quantum chemistry, theoretical physics, and combinatorics. We begin with the case $\mathbb{K}$P∞, and then identify the truncation required for passage to finite n. The calculations rely upon a ladder of long exact cohomology sequences, which compares cofibrations associated to the diagonals of the symmetric square and the corresponding Borel construction. These incorporate the one-point compactifications of classic configuration spaces of unordered pairs of points in $\mathbb{K}$Pn, which are identified as Thom spaces by combining Löwdin's symmetric orthogonalisation (and its quaternionic analogue) with a dash of Pin geometry. The relations in the ensuing cohomology rings are conveniently expressed using generalised Fibonacci polynomials. Our conclusions are compatible with those of Gugnin mod torsion and Nakaoka mod 2, and with homological results of Milgram.

  • 11 - Nontrees

  • Charles R. Johnson, College of William and Mary, Virginia, Carlos M. Saiago, Universidade Nova de Lisboa, Portugal
  • Book:
    Eigenvalues, Multiplicities and Graphs
    Published online:
    09 February 2018
    Print publication:
    12 February 2018, pp 211-231

  • Summary

    Introduction and Observations

    The problem of multiplicity lists for Hermitian matrices is remarkably different when the underlying graph G is not a tree. Some highly algebraic aspects of the problem are the same. For example,

  • • Maximum multiplicity in H(G) corresponds to minimum rank in H(G).

  • • Multiplicity can change by at most 1 when a vertex is removed.

  • • Multiplicity can change under perturbation by at most the rank of the perturbation.

  • • Many of the results on perturbation of diagonal entries (Section 4.1) are the same.

    • However, as seen in Chapter 2, there is little vestige of the technology of Parter vertices for general graphs, and it maywell be that all vertices are downers, even when multiple eigenvalues are present. Two of the construction techniques we have described in Chapter 7 are almost not available, and the third, the IFT, is more problematic to carry out. The similarities above are too weak to be very helpful, and minimum rank for general graphs is known to be very subtle, while it has a lovely answer for trees. Thus, determination of possible multiplicities is even more difficult for general graphs and often involves looking at very particular cases.

    There are some things that have been said, or can be said, and we review a selection of them here.

    The Complete Graph

    The complete graph Kn on n ≥ 2 vertices is relatively easy to analyze, as it allows many multiplicity lists. However, it requires different techniques to verify that various spectra occur, and these techniques are useful for other dense graphs that we pursue in Section 11.7. One list that Kn obviously does not allow is just one eigenvalue of multiplicity n. An Hermitian matrix with this list would be similar to, and thus, equal to, a multiple of I. It would be diagonal and thus is allowed only by the trivial graph with n vertices and no edges.

Page 188 of 1251