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Secant varieties and degrees of invariants. (English) Zbl 1440.13032

Let \(G\) be a connected complex reductive algebraic group and \(V\) be a finite dimensional \(G\)-module. In the paper under review, the author studies the ring of invariant polynomials \(\mathbb{C}[V]^{G}\) and in particular the degrees of the generators of \(\mathbb{C}[V]^{G}\) in a minimal generating set. The first main result is Theorem 1, which identifies certain divisors of the degrees of the generators of \(\mathbb{C}[V]^{G}\). In more detail, let \(H \subset G\) be a Cartan subgroup with root system \(\Delta\) and let \(\Lambda(V)\) denote the set of \(H\)-weights of \(V\). Suppose that \(M\) is a subset of \(\Lambda(V)\) which satisfies the following two conditions:
(i)
\(M \cap (M + \Delta) = \emptyset\) (such sets are called root-distinct)
(ii)
\(M\) is linearly dependent over \(\mathbb{Z}_{>0}\) and minimal with this property.
Let \(b_M =\sum_{ \nu \in M} b_{\nu}\), where \(b_{\nu} \in \mathbb{Z}_{>0}\) are the unique coefficients with greatest common divisor equal to \(1\) such that \(\sum_{\nu \in M} b_{\nu}\nu = 0\). Then, Theorem 1 states that \(\mathbb{C}[V]^{G}\) admits a generator of degree \(kb_M\) for some integer \(k\geq 1\). A similar construction of invariants for the case of tori has been introduced in D. Wehlau [Ann. Inst. Fourier, Vol. 43, 1055–1066 (1993, Zbl 0789.14009)].
Theorem 1 is proven in Section 2 along with some properties of root-distinct sets which follow N. Wildberger [Trans. AMS 330, 257–268 (1992, Zbl 0762.22012)].
In Sections 3 and 4, the author considers the case when \(G\) is semisimple and \(V = V(\lambda)\) is an irreducible representation of \(G\) with highest weight \(\lambda \neq 0\). Then there is a unique closed \(G\)-orbit in \(\mathbb{P}(V)\), namely \(\mathbb{X} = G[v_{\lambda}]\), where \(v_{\lambda}\) is a highest weight vector corresponding to \(\lambda\). The main result of Section 3 is Theorem 10, which gives a lower bound for the degrees of the generators of \(\mathbb{C}[V]^{G}\) using the secant varieties of \(\mathbb{X}\). More precisely, let \(J = \mathbb{C}[V]^{G}_{\geq 1}\) denote the ideal in the invariant ring vanishing at 0 and let \(\mathbb{P}^{\mathrm{us}} \subset \mathbb{P}\) be the zero-locus of \(J\). The complement \(\mathbb{P}^{\mathrm{ss}} = \mathbb{P} \setminus \mathbb{P}^{\mathrm{us}}\) is called the semistable locus. By \(\Sigma_r = \sigma_r(\mathbb{X})\) the author denotes the \(r\)-th secant variety of \(\mathbb{X}\). The rank of semistability of \(V\) (which exists when \(\mathbb{C}[V]^G \neq \mathbb{C}\)) is defined by \[ r_{\mathrm{ss}} = \min \{r \in \mathbb{N}; \Sigma_r \cap \mathbb{P}^{\mathrm{ss}}\neq \emptyset \}. \] Then, Theorem 10 states that if \(\mathbb{C}[V]^G \neq \mathbb{C}\) and if \(d_1\) denotes the minimal positive degree of a generator of \(\mathbb{C}[V]^G\), then \(r_{\mathrm{ss}} \leq d_1\). The author also gives examples that this lower bound may or may not be exact.
In Section 4, the author considers a special class of projective varieties \(\mathbb{X} \subset \mathbb{P}(V)\), called rs-continuous. For this class of varieties there is a bijective correspondence between the degrees of the generators \(\{d_1, \dots, \mathrm{No}(G,V)\}\) of \(\mathbb{C}[V]^G\) in a minimal set of generators, where \(\mathrm{No}(G,V)\) denotes the Noether number, i.e., the maximal degree of a generator of \(\mathbb{C}[V]^G\), and the set of numbers \(\{r_{\mathrm{ss}}, \dots, r_{g}\}\), where \(\Sigma_{r_{\mathrm{ss}}}, \dots, \Sigma_{r_g} = \mathbb{P}(V)\) are the secant varieties of \(\mathbb{X}\), which intersect the semistable locus \(\mathbb{P}^{\mathrm{ss}}\).

MSC:

13A50 Actions of groups on commutative rings; invariant theory
14L24 Geometric invariant theory
14M15 Grassmannians, Schubert varieties, flag manifolds
14N07 Secant varieties, tensor rank, varieties of sums of powers
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
22E46 Semisimple Lie groups and their representations
53D20 Momentum maps; symplectic reduction
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