A survey of the additive eigenvalue problem (with Appendix by M. Kapovich). (English) Zbl 1354.14074
This is a survey article on the classical Hermitian eigenvalue problem and its generalization to an arbitrary connected reductive group. The classical Hermitian eigenvalue problem starting with the work of H. Weyl [Math. Ann. 71, 441–479 (1912; JFM 43.0436.01)] is the following: determines all triples of \(n\)-tuples of nonincreasing real numbers: \(\lambda=(\lambda_1,\dots,\lambda_n)\), \(\mu=(\mu_1,\dots,\mu_n)\) et \(\nu=(\nu_1,\dots,\nu_n)\) such that there exist Hermitian matrices \(A\), \(B\), \(C=A+B\) with eigenvalues \(\lambda\), \(\mu\) et \(\nu\).
The solution was conjecture given by A. Horn [Pac. J. Math. 12, 225–241 (1962; Zbl 0112.01501)] and proved by A. A. Klyachko [Sel. Math., New Ser. 4, No. 3, 419–445 (1998; Zbl 0915.14010)] and A. Knutson and T. Tao [J. Am. Math. Soc. 12, No. 4, 1055–1090 (1999; Zbl 0944.05097)] The sets \(\lambda\), \(\mu\) et \(\nu\) are the solutions of the inequalities: \(\sum^{n}_{i=1}\nu_i =\sum^{n}_{i=1}\lambda_i+\sum^{n}_{i=1}\mu_i\) et \(\sum_{k\in K}\nu_i \leq\sum_{i\in I}\lambda_j+\sum_{j\in J}\mu_i\) for an appropriate family of sets \((I,J,K)\). It was proved by P. Belkale [Compos. Math. 129, No. 1, 67–86 (2001; Zbl 1042.14031)] and A. Knutson et al. [J. Am. Math. Soc. 17, No. 1, 19–48 (2004; Zbl 1043.05111)] that the above system of inequalities is overdetermined.
The above Hermitian eigenvalue problem has been generalized to an arbitrary complex semisimple group. Let \(G\) be a connected, semisimple complex algebraic group. We fix a Borel subgroup, a maximal torus \(H\), and a maximal compact subgroup \(K\) and let \(\underline{h}\) and \(\underline{k}\) be their Lie algebras. Let \(\underline{h}_+\subset \underline{h}\) be the positive Weyl chamber. There is a natural homeomorphism \(f : \underline{k}/K \rightarrow \underline{h}_+\). For any positive integer \(s\), define the eigencone \[ \overline{\Gamma}_s(\underline{g})=\{(h_1,\dots,h_s)\in \underline{h}^s_+\mathrm{\;such \;that\;} \exists (k_1,\dots,k_s)\in \underline{k}^s \mathrm{\;with \;}\sum k_i=0\mathrm{\;et\;}f(k_i)=h_i\;\forall i\}. \] The aim of the general additive eigenvalue problem is to find the inequalities describing the eigencone \(\overline{\Gamma}(\underline{g})\) explicitly.
The general additive eigenvalue problem was solved by A. Berenstein and R. Sjamaar [J. Am. Math. Soc. 13, No. 2, 433–466 (2000; Zbl 0979.53092)], M. Kapovich et al. [Geom. Funct. Anal. 19, No. 4, 1081–1100 (2009; Zbl 1205.53037)] and P. Belkale and S. Kumar [Invent. Math. 166, No. 1, 185–228 (2006; Zbl 1106.14037)] using the deformed cup product (known as the Belkale-Kumar product) on singular homology with integral coefficients \(H^* (G/P , Z)\) of a flag variety \(G/P\).
Then, one can determine the saturated tensor semigroup \(\overline{\Gamma}(G)\). Similar to the eigencone, one defines the saturated tensor semigroup \[ \overline{\Gamma}_s(G)=\{(\lambda_1,\dots,\lambda_s)\mathrm{\;dominant\;characters\;such\;that\;} \exists\; \]
\[ [V(N(\lambda_1)\otimes\dots\otimes V(N(\lambda_s))]^G\neq0 \mathrm{\;for\;some \;}N\geq1\}. \] \(\overline{\Gamma}_s(G)\) is identified, via the Killing form, with the set of dominant characters in \(\overline{\Gamma}_s(\underline{g})\).
The proof of previous result follows from the Hilbert-Mumford criterion for semistability, Kempf’s maximally destabilizing one parameter subgroups, Kempf’s parabolic subgroups associated to unstable points and the notion of Levi-movability.
An “explicit” determination of the eigencone hinges upon understanding the Belkale-Kumar product in the Schubert basis, for all the maximal parabolic subgroups P. This product is easier to understand than the usual cup product since in general “many more” terms in the Belkale-Kumar product in the Schubert basis drop out. However, the Belkale-Kumar product has a drawback in that it is not functorial.
Finally the author consider the saturation problem which aims at connecting the tensor product semigroup \[ T_s(G)= \{(\lambda_1,\dots,\lambda_s) \mathrm{\;dominant\;weights\;such \;that\;} [V (\lambda_1)\otimes \dots\otimes V(\lambda_s )]^G\neq0 \} \] with the saturated tensor product semigroup \(\Gamma_s (G)\). An integer \(d \geq 1\) is called a saturation factor for \(G\), if for any \((\lambda,\mu,\nu) \in \Gamma_3(G)\) such that \(\lambda+\mu+\nu\) belongs to the root lattice, then \((d\lambda,d\mu,d\nu)\in T_3 (G)\). Such a \(d\) always exists. If \(d = 1\) is a saturation factor for \(G\), we say that the saturation property holds for \(G\).
The saturation theorem of A. Knutson and T. Tao [J. Am. Math. Soc. 12, No. 4, 1055–1090 (1999; Zbl 0944.05097)] asserts that the saturation property holds for \(G =\mathrm{SL}(n)\). A general result (though not optimal) on saturation factor is obtained by M. Kapovich and J. J. Millson [Groups Geom. Dyn. 2, No. 3, 405–480 (2008; Zbl 1147.22011)].
The solution was conjecture given by A. Horn [Pac. J. Math. 12, 225–241 (1962; Zbl 0112.01501)] and proved by A. A. Klyachko [Sel. Math., New Ser. 4, No. 3, 419–445 (1998; Zbl 0915.14010)] and A. Knutson and T. Tao [J. Am. Math. Soc. 12, No. 4, 1055–1090 (1999; Zbl 0944.05097)] The sets \(\lambda\), \(\mu\) et \(\nu\) are the solutions of the inequalities: \(\sum^{n}_{i=1}\nu_i =\sum^{n}_{i=1}\lambda_i+\sum^{n}_{i=1}\mu_i\) et \(\sum_{k\in K}\nu_i \leq\sum_{i\in I}\lambda_j+\sum_{j\in J}\mu_i\) for an appropriate family of sets \((I,J,K)\). It was proved by P. Belkale [Compos. Math. 129, No. 1, 67–86 (2001; Zbl 1042.14031)] and A. Knutson et al. [J. Am. Math. Soc. 17, No. 1, 19–48 (2004; Zbl 1043.05111)] that the above system of inequalities is overdetermined.
The above Hermitian eigenvalue problem has been generalized to an arbitrary complex semisimple group. Let \(G\) be a connected, semisimple complex algebraic group. We fix a Borel subgroup, a maximal torus \(H\), and a maximal compact subgroup \(K\) and let \(\underline{h}\) and \(\underline{k}\) be their Lie algebras. Let \(\underline{h}_+\subset \underline{h}\) be the positive Weyl chamber. There is a natural homeomorphism \(f : \underline{k}/K \rightarrow \underline{h}_+\). For any positive integer \(s\), define the eigencone \[ \overline{\Gamma}_s(\underline{g})=\{(h_1,\dots,h_s)\in \underline{h}^s_+\mathrm{\;such \;that\;} \exists (k_1,\dots,k_s)\in \underline{k}^s \mathrm{\;with \;}\sum k_i=0\mathrm{\;et\;}f(k_i)=h_i\;\forall i\}. \] The aim of the general additive eigenvalue problem is to find the inequalities describing the eigencone \(\overline{\Gamma}(\underline{g})\) explicitly.
The general additive eigenvalue problem was solved by A. Berenstein and R. Sjamaar [J. Am. Math. Soc. 13, No. 2, 433–466 (2000; Zbl 0979.53092)], M. Kapovich et al. [Geom. Funct. Anal. 19, No. 4, 1081–1100 (2009; Zbl 1205.53037)] and P. Belkale and S. Kumar [Invent. Math. 166, No. 1, 185–228 (2006; Zbl 1106.14037)] using the deformed cup product (known as the Belkale-Kumar product) on singular homology with integral coefficients \(H^* (G/P , Z)\) of a flag variety \(G/P\).
Then, one can determine the saturated tensor semigroup \(\overline{\Gamma}(G)\). Similar to the eigencone, one defines the saturated tensor semigroup \[ \overline{\Gamma}_s(G)=\{(\lambda_1,\dots,\lambda_s)\mathrm{\;dominant\;characters\;such\;that\;} \exists\; \]
\[ [V(N(\lambda_1)\otimes\dots\otimes V(N(\lambda_s))]^G\neq0 \mathrm{\;for\;some \;}N\geq1\}. \] \(\overline{\Gamma}_s(G)\) is identified, via the Killing form, with the set of dominant characters in \(\overline{\Gamma}_s(\underline{g})\).
The proof of previous result follows from the Hilbert-Mumford criterion for semistability, Kempf’s maximally destabilizing one parameter subgroups, Kempf’s parabolic subgroups associated to unstable points and the notion of Levi-movability.
An “explicit” determination of the eigencone hinges upon understanding the Belkale-Kumar product in the Schubert basis, for all the maximal parabolic subgroups P. This product is easier to understand than the usual cup product since in general “many more” terms in the Belkale-Kumar product in the Schubert basis drop out. However, the Belkale-Kumar product has a drawback in that it is not functorial.
Finally the author consider the saturation problem which aims at connecting the tensor product semigroup \[ T_s(G)= \{(\lambda_1,\dots,\lambda_s) \mathrm{\;dominant\;weights\;such \;that\;} [V (\lambda_1)\otimes \dots\otimes V(\lambda_s )]^G\neq0 \} \] with the saturated tensor product semigroup \(\Gamma_s (G)\). An integer \(d \geq 1\) is called a saturation factor for \(G\), if for any \((\lambda,\mu,\nu) \in \Gamma_3(G)\) such that \(\lambda+\mu+\nu\) belongs to the root lattice, then \((d\lambda,d\mu,d\nu)\in T_3 (G)\). Such a \(d\) always exists. If \(d = 1\) is a saturation factor for \(G\), we say that the saturation property holds for \(G\).
The saturation theorem of A. Knutson and T. Tao [J. Am. Math. Soc. 12, No. 4, 1055–1090 (1999; Zbl 0944.05097)] asserts that the saturation property holds for \(G =\mathrm{SL}(n)\). A general result (though not optimal) on saturation factor is obtained by M. Kapovich and J. J. Millson [Groups Geom. Dyn. 2, No. 3, 405–480 (2008; Zbl 1147.22011)].
Reviewer: Alessandro Ruzzi (Roma)
MSC:
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 20G15 | Linear algebraic groups over arbitrary fields |
Citations:
Zbl 0112.01501; Zbl 0915.14010; Zbl 0944.05097; Zbl 1042.14031; Zbl 1043.05111; Zbl 0979.53092; Zbl 1205.53037; Zbl 1106.14037; Zbl 1147.22011; JFM 43.0436.01References:
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