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2
votes
0answers
103 views

Quaternionic projective bundle in complex Grassmann bundle

"What is the fundamental class of the projective bundle of lines of a quaternionic bundle in the Grassmann bundle of 2-planes of the underlying complex bundle?" In Quaternionic projective space in ...
4
votes
1answer
205 views

Quaternionic projective space in complex Grassmannian

I would like to consider the quaternionic projective space $\mathbb{PH}^{n-1}\subset\mathbb{G}_2(\mathbb{C}^{2n})$ as a subvariety of the Grassmannian of complex 2-planes. For a real vector $e\in\...
1
vote
0answers
114 views

On the maximal powers of $q$ which arise in a quantum product

Let $X=G/P$ be a generalized flag variety (where $G$ denotes a connected, simply connected, semisimple complex linear algebraic group and $P$ a parabolic subgroup). In this paper by Fulton and ...
1
vote
0answers
76 views

Rank diagrams of permutations $w \in S_{m}$ in the study of complete flag varieties [closed]

I'm looking for some good references that may either prove or help to prove the following statement: Show that a matrix $r=(r_{pq})_{1 \leq p,q \leq m}$ defines a rank diagram for some pair of ...
9
votes
1answer
115 views

Cross between the nil-Hecke ring and the group ring of a Coxeter group

A Coxeter system $(W,S)$ has a set of generators $S=\{s_1,s_2,\ldots\}$ and the Coxeter group $W$ is determined by relations of the form $(s_is_j)^{m_{ij}}=1$ for some integers $m_{ij}$, where $m_{ii}=...
5
votes
0answers
116 views

Littlewood-Richardson rule for the complete flag variety: GapP complete?

The cohomology ring of a complete flag variety $X$ has a basis of Schubert classes $S_u$ for permutations $u$. Define the Littlewood-Richardson coefficient $c_{uv}^w$ for permutations $u,v,w$ to be ...
3
votes
0answers
49 views

Antichains defining facets of a certain cone

Let $(P,<)$ be a finite poset. Let $V$ be the free $\mathbb{R}$-vector space on $P \times \{0,1\}$; I'll write elements as sums of pairs of the form $(p,0)$ and $(0,q)$, so a general element is $$v ...
0
votes
1answer
125 views

expressing in terms of sum of (double) schubert polynomial

It is well known that Schubert polynomials form a basis for the polynomial ring $\mathbb{Z}[x_1,x_2,x_3,...]$. I am interested in knowing how to express a particular polynomial into sum of Schubert ...
12
votes
0answers
149 views

Coefficients of universal Schubert polynomials

Let $e_i^j$ be the elementary symmetric polynomial in $x_1,x_2,\ldots,x_j$. Then the ordinary Schubert polynomial has an expansion of the form $$S_u(x)=\sum_{i_1,i_2,\ldots,i_n}{a^{i_1,i_2,\ldots,i_n}...
1
vote
0answers
115 views

Schubert Calculus for the Full Flags

Almost all introductory texts on Schubert calculus discuss the Grassmannian case only. Does there exist a nice discussion of the full flag manifold case $SU(N)/T^{N-1}$? A low dimensional example ...
5
votes
1answer
86 views

Reference for restriction formula in terms of double Schubert polynomials

Everyone (that is, everyone who cares) knows that double Schubert polynomials represent Schubert classes in equivariant cohomology in type $A$. We also know that we can restrict Schubert classes to ...
9
votes
1answer
224 views

Reduction formula for Schubert polynomials

In my endless fiddling with formulas I discovered one that fills in the blanks in a generic formula I saw in a paper, but I'm wondering if maybe it's already known and the paper was just mentioning ...
8
votes
1answer
190 views

Most computationally efficient Littlewood-Richardson rule

There are many, many different versions of the Littlewood-Richardson rule: the original characterization via Yamanouchi words, Remmel's version, a description via the Poirier-Reutenauer bialgebra, the ...
11
votes
0answers
380 views

Product of a Schubert polynomial and a double Schubert polynomial

Let $S_u(x)$ be a Schubert polynomial and let $S_v(x;y)$ be a double Schubert polynomial. Then their product can be expressed in terms of the double Schubert polynomials as $$S_u(x)S_v(x;y)=\sum_w{c_{...
1
vote
1answer
119 views

Schubert Polynomials for Complex Projective Space

The Borel picture of the cohomology ring of a flag variety gives a description as a coset space, without identifying any representatives for the classes. Lascoux and Schützenberger gave specific ...
0
votes
1answer
314 views

cup-length of the first Chern class of complex grassmannian

Let $G_2(\mathbb{C}^{n+1})$ be the complex grassmannian. Then the cohomology ring $H^*(G_2(\mathbb{C}^{n+1});\mathbb{C})=\mathbb{C}[c_1,c_2]/(f_n,f_{n+1})$, where $f_k=\sum_{i=0}^{[k/2]}(-1)^{k-i}{{...
5
votes
1answer
213 views

Combinatorial proof of the Cauchy identity for double Schubert polynomials

The Cauchy identity for double Schubert polynomials states $$ \mathfrak{S}_w(x;-y) = \sum_{\substack{u,v \in S_n \\ w=v^{-1}u \\ l(w) = l(v) + l(u)}} \mathfrak{S}_u(x)\mathfrak{S}_v(y).$$ Is there a ...
4
votes
0answers
248 views

Littlewood-Richardson rule for Schubert polynomials

What is the current state of the problem of finding a combinatorial rule for multiplying two Schubert polynomials? Is the problem still open?
2
votes
1answer
117 views

Depth of Schubert cycles

For $a:a_1\geq \cdots\geq a_c$, let $\sigma_a$ be the corresponding Schubert cycle over $Gr(c,\infty)$. We say $a$ is of depth $k$ if $a_1-a_c=k$ ($c>1$). Let $a$ and $b$ be of depth $k_1$ and $k_2$...
4
votes
2answers
213 views

Upper bound for the product of Schubert cycles

Let $Gr(c,\infty)$ be the complex grassmannian of $c$-dimensional subspaces of the infinite dimensional complex space. Every finite dimensional grassmannian, $Gr(c,N)$, can be thought as a subspace of ...
10
votes
1answer
503 views

Fomin-Kirillov algebras and Schubert calculus

In Fomin, Sergey; Kirillov, Anatol N. Quadratic algebras, Dunkl elements, and Schubert calculus. Advances in geometry, 147--182, Progr. Math., 172, Birkhäuser Boston, Boston, MA, 1999. MR1667680 (...
20
votes
0answers
570 views

Combinatorics of Quantum Schubert Polynomials

Let $S_n$ be the symmetric group. Let $s_i$ denote the adjacent transposition $(i \ i+1)$. For any permutation $w\in S_n$, an expression $w=s_{i_1}s_{i_2}\cdots s_{i_p}$ of minimal possible length is ...
2
votes
0answers
127 views

Computing intersection of cycles on the product of Grassmannians/Deligne-Lusztig varieties

My collaborators and I are preparing an interesting manuscript where the computation leads to something related to what we believe to be in the area of Schubert calculus; but none of us knows much ...
3
votes
2answers
252 views

Where can I look up some Schubert calculus numbers?

I don't know much about Schubert calculus, but I would like to know all of the intersection numbers $$ \#(X_{w_1} \cap X_{w_2} \cap X_{w_3}) $$ where $X_w$ indicates a Schubert variety (maybe ...
29
votes
2answers
1k views

Schubert calculus, as lowbrow as possible

Starting in a week I'm going to be an instructor at a summer program for exceptionally mathematically talented high school students, and I'm going to be teaching a class on Schubert calculus. The ...
6
votes
0answers
285 views

Real schubert calculus

Studying real enumerative questions I noticed, that the even-even Schubert varieties of the real Grassmannian $Gr:=Gr_{2k}(2n,\mathbb R)$ behave analogously to the complex case. I call a partition ...
8
votes
4answers
640 views

Subspaces of End(V) that can fix any vector

Suppose V is a finite-dimensional vector space and I have a linear subspace of its endomorphisms $$W \subseteq \mbox{End}(V).$$ How can I easily check if every vector of $V$ is fixed by some element ...