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Quantum Generalization of the Horn Conjecture
| Content Provider | Semantic Scholar |
|---|---|
| Author | Belkale, Prakash |
| Copyright Year | 2004 |
| Abstract | Consider the following additive eigenvalue problem for the special unitary group SU(n): • Characterize the possible eigenvalues (α, β, γ) of traceless Hermitian n× n matrices A, B and C which satisfy A + B + C = 0. (Recall that the Lie algebra of the special unitary group SU(n) is isomorphic to the real vector space of traceless Hermitian matrices as representations of SU(n) and hence the terminology “additive eigenvalue problem” for SU(n).) In 1962, Horn [17] gave a conjectural solution to a problem equivalent to the above additive eigenvalue problem for SU(n), by a recursively determined system of inequalities. In [19], Klyachko gave a solution to the additive eigenvalue problem for SU(n) in terms of a certain system of inequalities. To write down this system, we need to know which structure constants in the cohomology of Gr(r, n) (written in the Schubert basis) 0 < r < n are non-zero, where Gr(r, n) is the Grassmannian of r-dimensional vector subspaces of C. By the work of Klyachko and the saturation theorem of Knutson and Tao [20], the problem of determining whether a given structure constant in the cohomology of a Grassmannian Gr(r, n) (again in the Schubert basis) is non-zero is related to the additive eigenvalue problem for the smaller group SU(r). Horn’s original conjecture followed from these works of Klyachko and of Knutson and Tao. We refer the reader to Fulton’s survey article [12] for details. Our aim in this paper is to formulate and prove an analogue of Horn’s conjecture for the following multiplicative eigenvalue problem for the group SU(n). Let s ≥ 2 be a positive integer. • Characterize the possible eigenvalues (α, α, . . . , α) of matrices A, A, . . . , A ∈ SU(n) which satisfy A A · · ·A = 1. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.ams.org/journals/jams/2008-21-02/S0894-0347-07-00584-X/S0894-0347-07-00584-X.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/math/0303013v4.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/math/0303013v3.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Analog Arabic numeral 0 Eigenvalue Generalization (Psychology) Integer (number) Need to know Nomenclature Positive integer Quantum Recursion Small Utility functions on indivisible goods |
| Content Type | Text |
| Resource Type | Article |
