Questions tagged [schubert-calculus]
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2
votes
1answer
134 views
Typo in a paper definition of Schubert cells?
In the paper "Quantum state transformations and the Schubert calculus" by Sumit Daftuar and Patrick Hayden (Annals of Physics 315 (2005) 80-122) on page 91, we have following notations:
$A_r$ denotes ...
8
votes
0answers
175 views
Does anyone know of this manifestation of the Littlewood-Richardson coefficients for the complete flag variety?
This is the culmination of about 11 years of research but after I discovered it I found a proof that was extremely trivial, so I'm wondering if it's already known.
Let $(a,b)$ with $a < b$ ...
10
votes
1answer
360 views
Proving Positivity for Schubert Calculus
In study of the cohomology ring of the Grassmannians, which is usually known as Schubert calculus, one usually deals with a distinguished basis known as the Schubert basis $\{\sigma_\lambda\}$. One of ...
1
vote
1answer
143 views
Coefficients of the monomials appearing in a Schubert polynomial
It is known that the coefficients of the monomials appearing in a Schubert polynomial are always positive. My question is: Is it always true that at least one such coefficient must be $1$? If that is ...
5
votes
0answers
134 views
Can we see the symmetry of the quantum Schubert polynomial of a point
Let $X=G/B$ be a homogeneous space and consider the quantization map
$$
S_W\otimes\mathbb{C}[q]\to(S(\mathfrak{h})\otimes\mathbb{C}[q])/I_W^q\,,
$$
where
$S_W$ is the coinvariant algebra of the Weyl ...
3
votes
0answers
168 views
T-equivariant homology of affine Grassmannian
Let $G=SL_n$, denote the affine Grassmannian $Gr:=Gr_{G}=\mathcal{G}/\mathcal{P}$, where $\mathcal{G}=G(\mathbb{C}((z)))$ and $\mathcal{P}=G(\mathbb{C}[[z]])$. We know that $R:=H_*^T(Gr)\cong H_*^T(\...
1
vote
0answers
100 views
intersect a subvariety with a Schubert variety
Let $Y$ be an irreducible subvariety inside $Gr(r,n)$ (Grassmannian of $r$-plane inside $\mathbb{C}^n$) and $X_\lambda$ be a Schubert variety corresponding to $\lambda$. Assume that $codim(Y)+codim(X_\...
6
votes
2answers
316 views
Some Elementary Schubert Calculus Calculations
Here are some simple geometry problems I am unable to resolve to my satisfaction. I asked the question on Math Stack (https://math.stackexchange.com/questions/2713754/a-problem-in-elementary-...
5
votes
0answers
165 views
Number of bitangents to connected algebraic curve
Schubert
showed that a plane algebraic curve of degree $d$ has
at most
$$
\tfrac{1}{2} d (d-2) (d-3) (d+2)
=
\tfrac{1}{2}d^4-d^3-\tfrac{9}{2} d^2 +9 d
$$
bitangents (a.k.a., double tangents).
And ...
11
votes
0answers
200 views
Question on a reduction in Kirillov's paper on positivity of divided difference operators
As the title says, my question is on a specific argument in Kirillov - Skew divided difference operators and Schubert polynomials (journal, MSN) on positivity of divided difference operators. I recall ...
0
votes
0answers
154 views
Number of Generators of the Cohomology Ring of the Grassmannians
For complex projective space, its cohomology ring has $1$ generator. Extending up to the first Grassmannian which is not a projective space, that is, Grass$(4,2)$, a direct investigation shows that it ...
1
vote
0answers
113 views
Algebra Invariants of Schubert Calculus
For the Grassmannian Gr[N,k] of $k$-planes in $\mathbb{C}^N$, the cohomology ring $H^*(Gr[N,k])$ is a much studied object in an area called Schubert calculus. As a complex algebra, $H^*(Gr[N,k])$ is ...
14
votes
3answers
420 views
Schubert calculus expressed in terms of the cotangent space of the Grassmannians
Let $T^*_{\mathbb{C}}(Gr_{n,r})$ denote the cotangent space of the Grassmannian of $r$-planes in $\mathbb{C}^n$. Moreover, let $\Lambda^\bullet$ denote the exterior algebra of $T^*_{\mathbb{C}}(Gr_{n,...
5
votes
0answers
107 views
Fubini--Study Orthogonality for Schubert Calculus
Consider the following points:
$\bullet$ Let ${\cal Harm}(n,d)$ denote the harmonic forms of the de Rham complex of the Grassmannian $Gr_{\mathbb{C}}(n,d)$ with respect to the Riemannian metric ...
10
votes
1answer
288 views
Concrete description of an exceptional minuscule variety
Let $G$ be a complete reductive Lie group. A simple root $\alpha$ is said to be minuscule if the multiplicity of the coroot $\alpha^\vee$ in $\beta^\vee$ is at most $1$ for all positive roots $\beta$. ...
1
vote
1answer
141 views
Counting cosets in the Quotient of Weyl groups
Let $\Sigma=G/P$ be a flag variety of type $A$,i.e. $G=SL_{n+1}$ and is a stabilizer of the partial flag $0 \subset V_{d_1}\subset \cdots\subset V_{d_r}\subset V_{n+1}$ of length $r$. Let $W$ be its ...
5
votes
2answers
167 views
Positivity of coefficients of a polynomial derived from Schubert polynomials
Let $W=\bigcup_{n=1}^\infty S_n$ be the union of all symmetric groups $S_n$. For an element $w\in W$, denote by $\mathfrak{S}_w$ the Schubert polynomial associated to $w$, and by $\partial_w$ the ...
6
votes
1answer
297 views
Combinatorics of the Cohomology Ring of the Lagrangian Grassmannians
The Lagrangian Grassmannian is an important example in symplectic geometry, see here or here for details. It shares many similarities with the ordinary Grassmannians (as one would expect from the name)...
11
votes
1answer
542 views
Expected number of lines meeting four given lines or “what is 1.72…”
Given four random lines in $\mathbb{R}P^3$, how many lines intersect all of those lines?
In the recent paper Probabilistic Schubert Calculus, Peter Bürgisser and Antonio Lerario
discuss this question ...
4
votes
1answer
414 views
Applications of Schubert calculus
Schubert calculus is a venerable field in mathematics where the object of study is the cohomology ring of the Grassmannians. Since it has been around for over a hundred years one might wonder if any ...
4
votes
0answers
122 views
Pushforwards of higher-rank vector bundles on flags
Let $V \cong \mathbb{C}^3$ and let $\pi: Fl(V) \to \mathbb{P}(V)$ be the projection from the flag variety to the projective space (of lines) of $V$. Let $L \subset H \subset \mathbb{C}^3$ be the ...
3
votes
0answers
147 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
357 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
130 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
82 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 ...
10
votes
1answer
146 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
163 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
52 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 $$v ...
0
votes
1answer
147 views
expressing in terms of sum of (double) schubert polynomial
It is well known that Schubert polynomials form a basis for the polynomial ring $\mathbb{Z}[x_1,x_2,x_3,...]$.
I am interested in knowing how to express a particular polynomial into sum of Schubert ...
12
votes
0answers
188 views
Coefficients of universal Schubert polynomials
Let $e_i^j$ be the elementary symmetric polynomial in $x_1,x_2,\ldots,x_j$. Then the ordinary Schubert polynomial has an expansion of the form
$$S_u(x)=\sum_{i_1,i_2,\ldots,i_n}{a^{i_1,i_2,\ldots,i_n}...
2
votes
0answers
166 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
111 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 ...
10
votes
1answer
305 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
258 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
513 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
$$S_u(x)S_v(x;y)=\sum_w{c_{...
1
vote
1answer
140 views
Schubert Polynomials for Complex Projective Space
The Borel picture of the cohomology ring of a flag variety gives a description as a coset space, without identifying any representatives for the classes. Lascoux and Schützenberger gave specific ...
1
vote
1answer
372 views
cup-length of the first Chern class of complex grassmannian
Let $G_2(\mathbb{C}^{n+1})$ be the complex grassmannian.
Then the cohomology ring $H^*(G_2(\mathbb{C}^{n+1});\mathbb{C})=\mathbb{C}[c_1,c_2]/(f_n,f_{n+1})$, where $f_k=\sum_{i=0}^{[k/2]}(-1)^{k-i}{{...
5
votes
1answer
307 views
Combinatorial proof of the Cauchy identity for double Schubert polynomials
The Cauchy identity for double Schubert polynomials states
$$ \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).$$
Is there a ...
5
votes
0answers
317 views
Littlewood-Richardson rule for Schubert polynomials
What is the current state of the problem of finding a combinatorial rule for multiplying two Schubert polynomials? Is the problem still open?
2
votes
1answer
131 views
Depth of Schubert cycles
For $a:a_1\geq \cdots\geq a_c$, let $\sigma_a$ be the corresponding Schubert cycle over $Gr(c,\infty)$. We say $a$ is of depth $k$ if $a_1-a_c=k$ ($c>1$). Let $a$ and $b$ be of depth $k_1$ and $k_2$...
4
votes
2answers
222 views
Upper bound for the product of Schubert cycles
Let $Gr(c,\infty)$ be the complex grassmannian of $c$-dimensional subspaces of the infinite dimensional complex space. Every finite dimensional grassmannian, $Gr(c,N)$, can be thought as a subspace of ...
11
votes
1answer
674 views
Fomin-Kirillov algebras and Schubert calculus
In
Fomin, Sergey; Kirillov, Anatol N. Quadratic algebras, Dunkl elements, and
Schubert calculus. Advances in geometry, 147--182, Progr. Math., 172,
Birkhäuser Boston, Boston, MA, 1999. MR1667680 (...
21
votes
0answers
695 views
Combinatorics of Quantum Schubert Polynomials
Let $S_n$ be the symmetric group. Let $s_i$ denote the adjacent transposition $(i \ i+1)$. For any permutation $w\in S_n$, an expression $w=s_{i_1}s_{i_2}\cdots s_{i_p}$ of minimal possible length is ...
3
votes
0answers
156 views
Computing intersection of cycles on the product of Grassmannians/Deligne-Lusztig varieties
My collaborators and I are preparing an interesting manuscript where the computation leads to something related to what we believe to be in the area of Schubert calculus; but none of us knows much ...
3
votes
2answers
339 views
Where can I look up some Schubert calculus numbers?
I don't know much about Schubert calculus, but I would like to know all of the intersection numbers
$$
\#(X_{w_1} \cap X_{w_2} \cap X_{w_3})
$$
where $X_w$ indicates a Schubert variety (maybe ...
31
votes
2answers
2k views
Schubert calculus, as lowbrow as possible
Starting in a week I'm going to be an instructor at a summer program for exceptionally mathematically talented high school students, and I'm going to be teaching a class on Schubert calculus. The ...
8
votes
1answer
446 views
Real schubert calculus
Studying real enumerative questions I noticed, that the even-even Schubert varieties of the real Grassmannian $Gr:=Gr_{2k}(2n,\mathbb R)$ behave analogously to the complex case.
I call a partition ...
8
votes
4answers
730 views
Subspaces of End(V) that can fix any vector
Suppose V is a finite-dimensional vector space and I have a linear subspace of its endomorphisms
$$W \subseteq \mbox{End}(V).$$
How can I easily check if every vector of $V$ is fixed by some element ...
