Eigenvalues of products of unitary matrices and quantum Schubert calculus.

*(English)*Zbl 1004.14013From the introduction: Beginning with Weyl, many mathematicians have been interested in the following question: given the eigenvalues of two Hermitian matrices, what are the possible eigenvalues of their sum? In a recent paper [Sel. Math., New Ser. 4, 419-445 (1998; Zbl 0915.14010)], A. A. Klyachko observed that a solution to this problem is given by an application of Mumford’s criterion in geometric invariant theory [see also the expository paper by W. Fulton in Astérisque 252, Sémin. Bourbaki 1997/98, Exp. No. 845, 255-269 (1998; Zbl 0929.15006)]. The eigenvalue inequalities are derived from products in Schubert calculus. In particular, Weyl’s inequalities correspond to Schubert calculus in projective space. The necessity of these conditions is due to U. Helmke and J. Rosenthal [Math. Nachr. 171, 207-225 (1995; Zbl 0815.15012)].

One of the fascinating points about the above problem is that there are several equivalent formulations noted by Klyachko. For instance, the problem is related to the following question in representation theory: Given a collection of irreducible representations of \(\text{SU}(n)\), which irreducibles appear in the tensor product? A second equivalent problem involves toric vector bundles over the complex projective plane.

In this paper we investigate the corresponding problem for products of unitary matrices. This question also has a relationship with a representation-theoretic problem, that of the decomposition of the fusion product of representations [see A. Knutson and T. C. Tao, J. Am. Math. Soc. 12, 1055-1090 (1999; Zbl 0944.05097)]. The solution to the multiplicative problem is also derived from geometric invariant theory, namely from the Mehta-Seshadri theory of parabolic bundles over the projective line. The main result of this paper shows that the eigenvalue inequalities are derived from products in quantum Schubert calculus. This improves a result of I. Biswas [Asian J. Math. 3, 333-344 (1999; Zbl 0982.14022)], who gave the first description of these inequalities. A similar result has been obtained independently by P. Belkale [Compos. Math. 129, 67-86 (2001; Zbl 1042.14031)].

The proof is an application of the Mehta-Seshadri theorem. A set of unitary matrices \(A_1,\dots, A_l\) such that each \(A_i\) lies in a conjugacy class \(C_i\) and such that their product is the identity is equivalent to a unitary representation of the fundamental group of the \(l\) times punctured sphere, with each generator \(\gamma_i\) being mapped to the conjugacy class \(C_i\). By the Mehta-Seshadri theorem such a representation exists if and only if there exists a semi-stable parabolic bundle on \(\mathbb{P}^1\) with \(l\) parabolic points whose parabolic weights come from the choice of conjugacy classes \(C_i\). This last interpretation of the original eigenvalue problem can be related to the Gromov-Witten invariants of the Grassmannian and is done in this paper.

Moreover, we investigate how factorization and hidden symmetries of these Gromov-Witten invariants relate to the multiplicative eigenvalue problem.

Finally we explain the representation-theoretic interpretation of \(\Delta_q(l)\) in terms of the Verlinde algebra of \(\text{SU}(n)\).

One of the fascinating points about the above problem is that there are several equivalent formulations noted by Klyachko. For instance, the problem is related to the following question in representation theory: Given a collection of irreducible representations of \(\text{SU}(n)\), which irreducibles appear in the tensor product? A second equivalent problem involves toric vector bundles over the complex projective plane.

In this paper we investigate the corresponding problem for products of unitary matrices. This question also has a relationship with a representation-theoretic problem, that of the decomposition of the fusion product of representations [see A. Knutson and T. C. Tao, J. Am. Math. Soc. 12, 1055-1090 (1999; Zbl 0944.05097)]. The solution to the multiplicative problem is also derived from geometric invariant theory, namely from the Mehta-Seshadri theory of parabolic bundles over the projective line. The main result of this paper shows that the eigenvalue inequalities are derived from products in quantum Schubert calculus. This improves a result of I. Biswas [Asian J. Math. 3, 333-344 (1999; Zbl 0982.14022)], who gave the first description of these inequalities. A similar result has been obtained independently by P. Belkale [Compos. Math. 129, 67-86 (2001; Zbl 1042.14031)].

The proof is an application of the Mehta-Seshadri theorem. A set of unitary matrices \(A_1,\dots, A_l\) such that each \(A_i\) lies in a conjugacy class \(C_i\) and such that their product is the identity is equivalent to a unitary representation of the fundamental group of the \(l\) times punctured sphere, with each generator \(\gamma_i\) being mapped to the conjugacy class \(C_i\). By the Mehta-Seshadri theorem such a representation exists if and only if there exists a semi-stable parabolic bundle on \(\mathbb{P}^1\) with \(l\) parabolic points whose parabolic weights come from the choice of conjugacy classes \(C_i\). This last interpretation of the original eigenvalue problem can be related to the Gromov-Witten invariants of the Grassmannian and is done in this paper.

Moreover, we investigate how factorization and hidden symmetries of these Gromov-Witten invariants relate to the multiplicative eigenvalue problem.

Finally we explain the representation-theoretic interpretation of \(\Delta_q(l)\) in terms of the Verlinde algebra of \(\text{SU}(n)\).

##### MSC:

14N15 | Classical problems, Schubert calculus |

14H60 | Vector bundles on curves and their moduli |

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

15A42 | Inequalities involving eigenvalues and eigenvectors |