Invariant valuations on quaternionic vector spaces (Q2894442)

The paper under review deals with the spaces of \(\mathrm{Sp}(n)-\), \(\mathrm{Sp}(n)*\mathrm{U}(n)-\) and \(\mathrm{Sp}(n)*\mathrm{Sp}(1)-\)invariant, translation-invariant, continuous convex valuations (i.e., finitely additive maps on the space of compact convex bodies) on the quaternionic vector space. The main results of the paper include some combinatorial dimension formulae involving Young diagrams and Schur polynomials. The paper begins with an introductory chapter which contains notation and the statement of the main results (Theorems 1.1 and 1.2, and 1.3). In chapter 2, the author collects some known factsssssss from the theory of convex valuation. The only new statement is Proposition 2.6, which is central in the proof of Theorems 1.2 and 1.3. In Chapter 3, some background on quaternionic vector spaces and quaternionic groups and on representation theory for the groups \(\mathrm{SU}(2)\) and \(\mathrm{GL}(n,\mathbb{C})\) are given. Chapter 4 provides some facts on invariants of the group \(\mathrm{Sp}(n)\). The proof of Theorem 1.1 is contained in Chapter 5 and uses several tools from the theory of convex valuations: normal cycle, Klain's embedding theorem and the Alesker-Fourier transform. Theorem 1.2 is proved in the same chapter. In Chapter 6, the author uses some computations for \(\mathrm{SU}(2)\)-representations to prove Theorem 1.3. Finally, in the appendix, the rather technical proof of a lemma which is used in Chapter 5 is given.