By Sara Sarason, V. Lakshmibai

"Singular Loci of Schubert forms" is a special paintings on the crossroads of illustration idea, algebraic geometry, and combinatorics. over the last two decades, many study articles were written at the topic in impressive journals. during this paintings, Billey and Lakshmibai have recreated and restructured a few of the theories and ways of these articles and current a clearer knowing of this significant subdiscipline of Schubert types – specifically singular loci. the main target, for this reason, is at the computations for the singular loci of Schubert forms and corresponding tangent areas. The equipment used comprise normal monomial conception, the nil Hecke ring, and Kazhdan-Lusztig concept. New effects are provided with adequate examples to stress key issues. A entire bibliography, index, and tables – the latter to not be stumbled on somewhere else within the arithmetic literature – around out this concise paintings. After a superb advent giving history fabric, the subjects are provided in a scientific model to have interaction a large readership of researchers and graduate students.

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Ad} t denotes the ordered d-tuple obtained from {al, ... , ad} by arranging its elements in ascending order. 5). Bjorner and Brenti [16] have shown that in fact one only needs to check those d for which d is a descent in (al ... an). 6. The flag variety SL(n)/ B. Let V = Kn. A sequence (0) = Va C VI C ... C Vn = V, such that dim Vi = i, is called a full flag in V. Let F(V) denote the set of all full flags in V. Let {ei' 1 ::; i ::; n} be the standard basis of Kn. The flag Fo = (Va c ... C Vi c ...

Description of T(w, id) In this section, we give a root system description of T(w, id) for G classical. 1. 1. Theorem. (ef. [98], [99]). Let f3 E R+. 1. Let G be of type An- Then f3 E N(w, id) ¢::::::} w 2: sf3. 2. Let G be of type en(a) If f3 = Ei - Eil or 2Ei, then f3 E N(w, id) ¢::::::} w 2: sf3. (b) If f3 = Ei +Ej, then f3 E N(w, id) ¢::::::} w 2: either S* w ~ either SE;+En • (c) Iff3=fi+fj,j w or SE; or ~ eithers Ei +E; SE;SE;+En. *

In particular, we have, for 1 ~ i,j ~ 2n, U(Eij) = -Ej'i', where Eij is the elementary matrix with 1 at the (i,j)-th place and 0 elsewhere. Further, Lie SO(2n) = {A E sl(2n) I E(tA)E = -A}. The Chevalley basis for Lie SO(2n) may be given as follows: H