Download e-book for kindle: Singular Loci of Schubert Varieties by Sara Sarason, V. Lakshmibai

By Sara Sarason, V. Lakshmibai

ISBN-10: 146121324X

ISBN-13: 9781461213246

ISBN-10: 1461270944

ISBN-13: 9781461270942

"Singular Loci of Schubert types" is a distinct paintings on the crossroads of illustration idea, algebraic geometry, and combinatorics. over the last twenty years, many learn articles were written at the topic in remarkable journals. during this paintings, Billey and Lakshmibai have recreated and restructured many of the theories and methods of these articles and current a clearer figuring out of this crucial subdiscipline of Schubert types – particularly singular loci. the focus, for that reason, is at the computations for the singular loci of Schubert forms and corresponding tangent areas. The tools used contain ordinary monomial conception, the nil Hecke ring, and Kazhdan-Lusztig idea. New effects are provided with adequate examples to stress key issues. A accomplished bibliography, index, and tables – the latter to not be stumbled on in different places within the arithmetic literature – around out this concise paintings. After a very good creation giving heritage fabric, the subjects are provided in a scientific style to interact a large readership of researchers and graduate students.

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Example text

A) If f3 = Ei - Ej, En, or Ei + En, then f3 E N(w,id) ¢::::::} w 2: sf3. 3. DESCRIPTION OF T(w,id) = (b) If f3 fi,i < n, then f3 E N(w,id) 53 -¢::::::> w ~ either SE;+En • (c) Iff3=fi+fj,jw or SE; or ~ eithers Ei +E; SE;SE;+En. 4. Let G be of type Dn. (a) Iff3=fk-fl, orfi+fj,j=n-l,n, thenf3EN(w,id) w ~ sf3. (b) If f3 = fi + fj, SE;+E; or j < n - 1, then f3 E N(w, id) -¢::::::> -¢::::::> w ~ either SEi-En SEi+En SE;+En_l. PROOF. 1, f3 E N(w, id) if and only if X- f3 qid(d) belongs to VW,Wd for all 1 ~ d ~ l.

The minimal and maximal representatives as permutations. Let W E W, and let wQin be the element in WQin that represents the coset wWQ. Under the identification of WQin with Ial, ... ,ak' let i = (il , ... ,ik ) be the element in Iah ... ,ak that corresponds to wQin. As a permutation, the element wQin is given by iI' followed by i2 \ i l arranged in ascending order, and so on, ending with {I, ... ,n} \ i k arranged in ascending order. Similarly, if wQax is the element in wQax that represents the coset wWQ , then as a permutation, the element wQax is given by i l arranged in descending order, followed by i2 \ i l arranged in descending order, etc.

All maximal chains have the same length, namely l(w) + 1; here, by a chain we mean a totally ordered subset of lid, w]. Consider an edge T' --+ T, where X (T') is a Schubert divisor in X(T), and give it the weight m(T, T'). Then degXQ(w) is simply the number of maximal chains with the edges counted with the respective multiplicities. To be very precise, to a maximal chain f : {w = ¢o > ¢Y! > '" > ¢r = id}, where r = l(w) = dim(X(w)), assign the weight n(f) := n~=1 m(¢i-l, ¢i)' Then degXQ(w) = L n(f), where the summation runs over all the maximal chains f in lid, w].

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Singular Loci of Schubert Varieties by Sara Sarason, V. Lakshmibai


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