Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mccme.ru/~smirnoff/papers/oberwolfach11.pdf Äàòà èçìåíåíèÿ: Sat Jan 16 07:43:29 2016 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 13:00:31 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: arp 220

Convex chains for Schub ert varieties Valentina Kiritchenko (joint work with Evgeny Smirnov and Vladlen Timorin) In [4], we constructed generalized Newton polytopes for Schubert subvarieties in the variety of complete flags in Cn . Every such "polytope" is a union of faces of a Gelfand­Zetlin polytope (the latter is a well-known Newton­Okounkov body for the flag variety). These unions of faces are responsible for Demazure characters of Schubert varieties and were originally used for Schubert calculus. The methods of [4] lead to an extension of Demazure (or divided difference) operators from representation theory and topology to the setting of convex geometry. Below I define divided difference operators acting on convex polytopes and outline some applications such as a simple inductive construction of Gelfand-Zetlin polytopes and their generalizations. The definition is based on the following observation. Let (µ, ) where µ, Zm denote the integer coordinate paral lelepiped {(x1 , . . . , xm )|µi xi i } m Rm , and let (x) for x Rm denote the sum of coordinates i=1 xi . Given a parallelepiped = (µ, ) Rm of dimension m - 1 (assume that µm = m ) and an integer C , there is a unique parallelepiped = (µ, ) Rm such that = {xm = µm } (that is, i = i for i < m) and t(x) ), () t(x) = DC (
xZd xZd

where DC is a Demazure-type operator on the ring Z[t, t-1 ] of Laurent polynomials in t: f - tf DC (f ) := , f := tC f (t-1 ). 1-t Indeed, an easy calculation (using the formula for the sum of a geometric progresm sion) shows that i=1 (µi + i ) = C which yields the value of m . Note that is a facet of unless = . We now use this observation in a more general context. A root space of rank n is a coordinate space Rd together with a direct sum decomposition Rd = R
d
1

. . . Rd

n

and a collection of linear functions l1 , . . . , ln (Rd ) such that li vanishes on Rdi . We always assume that the summands are coordinate subspaces so that Rd1 is spanned by the first d1 basis vectors etc. Let P Rd be a convex polytope in the root space. It is called a parapolytope if for all i = 1,. . . , n, the intersection of P with any parallel translate of Rdi is a coordinate parallelepiped. For instance, if d = n, that is, d1 = . . . = dn = 1, then every polytope is a parapolytope. For each i = 1,. . . , n, we now define a divided difference operator Ai on parapolytopes. In general, the operator Ai takes values in convex chains in Rd (see [3] for a definition) but in many cases of interest (see examples below) these convex chains will just be single convex parapolytopes.
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First, consider the case where P (c + Rdi ) for some c Rd , i.e. P = P (µ, ) is a coordinate parallelepiped. Here µ = (µ1 , . . . , µd ), = (1 , . . . , d ). Put Ni := d1 + . . . + di and N0 = 0. Assume that dim(P ) < di . Choose the smallest j [Ni-1 + 1, Ni ] such that µj = j . Define Ai (P ) to be the coordinate parallelepiped (µ, ), where k = k for all k = j and j is chosen so that
Ni k=Ni
-1

(µk + k ) = li (c), +1

()

that is, an analog of formula () holds for = P , = Ai (P ) and C = li (c). The definition yields a non-virtual coordinate parallelepiped if li (c) is sufficiently large and can be extended to other values of li (c) by linearity. For an arbitrary parapolytope P Rd define Ai (P ) as the union of Ai (P (c + di R )) over all c Rd : Ai (P ) = {Ai (P (c + Rdi ))}
cR
d

(assuming that dim(P (c + Rdi )) < di for all c Rd ). In other words, we first slice P by subspaces parallel to Rdi and then replace each slice with another parallelepiped according to (). Note that P is a facet of Ai (P ) unless Ai (P ) = P . It is easy to check that A2 = Ai (the same identity as for the classical Demazure i operators). Examples: (1) The simplest meaningful example is R2 = R R = {(x, y )} with the functions l1 = y and l2 = x. If P = (a, b) is a point, then A1 (P ) and A2 (P ) are segments: A1 (P ) = [(a, b), (b - a, b)],
1 2

A2 (P ) = [(a, b), (a, a - b)],

assuming that b a 2b. If b < 2a, then A1 (P ) is a virtual segment. If 2b > a, then A2 (P ) is virtual. If P = [(a, b), (a , b)] is a horizontal segment, then A2 (P ) is the trapezoid (or a skew trapezoid) with the vertices (a, b), (a , b), (a, a - b), (a , a - b). (2) A more interesting example is R3 = R2 R = {(x, y , z )} with the functions l1 = z and l2 = x+y . If P = [(a, b, c), (a , b, c)] is a segment in R2 , then A1 (P ) is the rectangle with the vertices (a, b, c), (a , b, c), (a, c - a - a - b, c), (a , c - a - a - b, c). Using this calculation and those in (1), it is easy to show that if P = (b, c, c) is a point and -b - c > b > c, then A1 A2 A1 (P ) is the 3-dimensional Gelfand­Zetlin polytope given by the inequalities a x b, b y c and x z y , where a + b + c = 0. (3) Generalizing the last example we now construct Gelfand­Zetlin polytopes - for arbitrary n via divided difference operators. For n N, put d = n(n2 1) . Consider the root space Rd = Rn-1 Rn-2 . . . R1 of rank (n - 1) with the functions li given by the formula: li (x) = i-1 (x) + i+1 (x). Here i (x) denotes the sum of those coordinates of x Rd that correspond to the subspace Rdi (put 0 = n = 0).


3 For every strictly dominant weight = (1 , . . . , n ) (that is, 1 > . . . > n ) of GLn such that 1 + . . . + n = 0, the Gelfand­Zetlin polytope Q coincides with the polytope [(A1 . . . An-1 )(A1 . . . An-2 ) . . . (A1 )] (p), d where p R is the point (2 , . . . , n ; 3 , . . . , n ; . . . ; n ). Similarly, divided difference operators for suitable root spaces allow one to construct the classical Gelfand­Zetlin polytopes for symplectic and orthogonal groups. They also yield an elementary description of more general string polytopes defined in [5] and might help to extend the results of [4] to arbitrary semisimple groups. As outlined below, these convex geometric operators are well suited for inductive constructions of Newton­Okounkov polytopes for line bundles on Bott towers and on Bott-Samelson varieties (for natural choice of a geometric valuation). The former polytopes were described in [2] and the latter are currently being computed by Dave Anderson. Bott towers. Consider a root space with d = n, that is, d1 = . . . = dn = 1. We have the decomposition Rn = R . . . R;
n

y = (y1 , . . . , yn )

into coordinate lines. Assume that the linear function li for i < n does not depend on y1 , . . . , yi , and ln = y1 . I can show that the polytope P := A1 . . . An (p) (for a point p Rn ) coincides with the Newton­Okounkov body for a Bott tower (that depends on l1 , . . . , ln ) together with a line bundle (that depends on p). For n = 2, a Bott tower is a Hirzebruch surface and P is a trapezoid (or a skew trapezoid) constructed similarly to the one in example (1). In general, a Bott tower is a toric variety obtained by successive pro jectivizations of rank two split vector bundles, and P is a multidimensional version of a trapezoid. Bott­Samelson resolutions. Let X = R(i1 , . . . , il ) be the Bott-Samelson variety corresponding to any sequence (i1 , . . . , il ) of roots of the group GLn . It can be obtained by successive pro jectivizations of rank two (usually non-split) vector bundles. Consider the root space Rd = Rd1 Rd2 . . . Rdn-1 with the functions li given by the formula li (x) = i-1 (x) + i+1 (x), where di is the number of times the root i occurs in the sequence (i1 , . . . , il ). Denote by Tv the parallel translation in the root space by a vector v Rd . Consider the polytope ] [ P = Ai1 Tv1 Ai2 . . . Tvl-1 Ail (p). In his talk, Dave Anderson described an algorithm for computing the Newton­ Okounkov body of a line bundle on X with respect to the valuation given by the flag of subvarieties {. . . R(il-1 , il ) R(il )}. Based on his computations for l = 3 [1], I conjecture that this Newton­Okounkov body coincides with P for suitable choice of a point p Rd and vectors vj Rdij for j = 1, . . . , l - 1. References
[1] Dave Anderson, Okounkov bodies and toric degenerations, preprint arXiv:1001.4566v2 [math.AG]


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[2] Michael Grossberg and Yael Karshon, Bott towers, complete integrability, and the extended character of representations, Duke Math. J. 76 (1994), no. 1 , 23­58. [3] A.G. Khovanskii, A.V. Pukhlikov, Finitely additive measures of virtual polytopes, St. Petersburg Mathematical Journal 4 (1993), no.2, 337­356 [4] Valentina Kiritchenko, Evgeny Smirnov, Vladlen Timorin, Schubert calculus and Gelfand­ Zetlin polytopes, preprint arXiv:1101.0278v2 [math.AG] [5] Peter Littelmann, Cones, crystals and patterns, Transformation Groups 3 (1998), pp. 145­ 179