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.
Read Online or Download Singular Loci of Schubert Varieties PDF
Similar combinatorics books
This revised and enlarged 5th variation positive aspects 4 new chapters, which include hugely unique and pleasant proofs for classics reminiscent of the spectral theorem from linear algebra, a few newer jewels just like the non-existence of the Borromean earrings and different surprises. From the Reviews". .. within PFTB (Proofs from The booklet) is certainly a glimpse of mathematical heaven, the place smart insights and gorgeous principles mix in outstanding and excellent methods.
Combinatorics and Algebraic Geometry have loved a fruitful interaction because the 19th century. Classical interactions comprise invariant concept, theta features and enumerative geometry. the purpose of this quantity is to introduce contemporary advancements in combinatorial algebraic geometry and to strategy algebraic geometry with a view in the direction of purposes, equivalent to tensor calculus and algebraic facts.
Finite Geometry and Combinatorial Applications
The projective and polar geometries that come up from a vector area over a finite box are fairly beneficial within the development of combinatorial gadgets, similar to latin squares, designs, codes and graphs. This e-book presents an advent to those geometries and their many purposes to different components of combinatorics.
- Applications of Combinatorics and Graph Theory to the Biological and Social Sciences
- Theory of Association Schemes
- Analytic combinatorics - symbolic combinatorics
- Combinatorics 86
- Combinatorics '90: Recent trends and applications
Extra info for Singular Loci of Schubert Varieties
Sample text
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
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