By I. N. Herstein

Starting summary Algebra with the vintage Herstein therapy.

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R denote simple roots then each (i, j) ∈ S corresponds to a positive root αi + αi+1 + . . + αj . Another useful interpretation is to regard a pair (i, j) ∈ S as a segment [i, j] = {i, i + 1, . . , j} in Z. A family m = mij ∈ ZS+ can be regarded as a collection of segments, containing mij copies of each [i, j]. Thus, elements of ZS+ can be called multisegments. The weight |m| of a multisegment m is defined as a sequence γ = {d1 , . . , dr ) ∈ Zr+ given by di = mkl for i = 1, . . , r. i∈[k,l] In other words, |m| records how many segments of m contain any given number i ∈ [1, r].

These embeddings can be done equivariantly. Points of V ∗ correspond to hyperplanes in Pn not passing through the origin 0 ∈ V ⊂ Pn . Suppose that O ⊂ V ∗ is a non-conical orbit. Then the dual ∗ ∗ variety O ⊂ Pn intersects with Cn non-trivially. Therefore, O ⊂ Pn is the ∗ ∗ closure of a conical variety in Cn . Therefore, its dual variety O ⊂ Pn ∗ does not intersect (Cn )∗ . But this contradicts the Reflexivity Theorem. 8. 9 ([Py]) Suppose that a connected algebraic group G acts linearly on a vector space V with a finite number of orbits.

We associate to m the multisegment m given by m = m − (i1 , j1 ) − (i2 , j2 ) − . . − (ip , jp ) + (i2 , j1 ) + (i3 , j2 ) + . . + (ip+1 , jp ), where we use the convention that (i, j) = 0 unless 1 ≤ i ≤ j ≤ r. 12 ([MW]) If the multisegment m is associated to m then ζ(m) = ζ(m ) + (i1 , ip ). The involution ζ can also be described in terms of irreducible finitedimensional representations of affine Hecke algebras and in terms of canonical bases for quantum groups, see [KZ]. 3 Parabolic Subgroups With Abelian Unipotent Radical Let L be a simple algebraic group and P ⊂ L a parabolic subgroup with abelian unipotent radical.