By Martin Lorenz
Multiplicative invariant thought, as a study region in its personal correct in the wider spectrum of invariant thought, is of rather fresh classic. the current textual content bargains a coherent account of the fundamental effects completed therefore far..
Multiplicative invariant thought is in detail tied to crucial representations of finite teams. for that reason, the sector has a predominantly discrete, algebraic taste. Geometry, in particular the speculation of algebraic teams, enters via Weyl teams and their root lattices in addition to through personality lattices of algebraic tori.
Throughout the textual content, quite a few specific examples of multiplicative invariant algebras and fields are offered, together with the entire checklist of all multiplicative invariant algebras for lattices of rank 2.
The ebook is meant for graduate and postgraduate scholars in addition to researchers in crucial illustration conception, commutative algebra and, normally, invariant theory.
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Extra info for Multiplicative Invariant Theory
N} of L, or a ﬁxed isomorphism L ∼ = n , gives rise to a -algebra isomorphism of [L] with ±1 ei → xi . Therefore, we may the Laurent polynomial algebra [x±1 1 , . . , xn ] via x m think of the representatives x ∈ [L] of lattice elements m ∈ L as monomials in ±1 x±1 1 , . . , xn . 2 Multiplicative Actions If L is a G-lattice for some group G then the action of G on L extends uniquely to an action by -algebra automorphisms on [L] via ✁ ✁ km xm ) = g( m∈L km xg(m) . 2) m∈L This type of action is called a multiplicative action.
Then g maps Φ to itself, since Φ is G-stable. Moreover, since g has order 2 , we have g(v) − v ∈ KerV (g + Id) = g for all v ∈ V . Thus, g(v) = v + rg,v g for some rg,v ∈ . The map v → rg,v is the ∗ required linear form ∨ g ∈V . For R3, note that, choosing v ∈ Φ in the preceding paragraph, we obtain g(v) − v ∈ KerL (g + Id) = g ; so ∨ g , v = rg,v ∈ , as required. − Finally, since L/L− g is -free, no element of 2L can be a generator of Lg . This proves R4, thereby completing the proof that Φ is a reduced root system in V .
N − 1). This map is clearly surjective for n ≥ 3 and it passes down to the symmetric square S2 An−1 . Equivariance for Sn can be checked by direct calculation. 19) r=s and the Sn -epimorphism (er ⊗ es ) er ⊗ es → es . Un , r=s The map ϕ is the restriction of the composite of these maps to A⊗2 n−1 . The desired sequence now follows by letting Kn denote the kernel of the epimorphism Un afforded by ϕ. Counting ranks, we see that K3 = 0. 19]. Thus, we must have ✝ ✆ ✝ ✝ ✆ ✝ Kn ⊗ ✝ ✆ ∼ = S (n−2,2) , which shows that Kn is rationally irreducible.