Questions tagged [schubert-calculus]

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Intersection cycle in a product of Grassmannians

Let $G(k,n)$ denote the Grassmiannian of $k$-planes in $\mathbb C^n$. Let's define $$ I_j =\{ (\Lambda_1,\Lambda_2 ) \in G(k,n) \times G(l,n) \, | \, \dim(\Lambda_1 \cap \Lambda_2) \geq j \}. $$ These ...
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1 answer
377 views

Cohomology ring of grassmannian and Pieri rule

I am sorry if this question is not for mathoverflow. I asked the same question on stackexchange (https://math.stackexchange.com/questions/4203667/cohomology-ring-of-grassmannian-and-pieri-rule), but I ...
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12 votes
1 answer
345 views

Planes in Lagrangian Grassmannians

Let $LG(h,2h)$ be the Lagrangian Grassmannian of subspaces of dimension $h$ of a complex vector space of dimension $2h$. For instace, $LG(1,2)=\mathbb{P}^1$, and $LG(2,4)\subset\mathbb{P}^4$ is a ...
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Flag variety as monoid and Schubert calculus

The lattice of linear subspaces in a vector space V can be provided with a structure of monoid by considering the subspace generated by the union of two subspaces as the monoid operation. When looking ...
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2 votes
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144 views

Quadrics tangent to lines

I think that the following must be a basic question in enumerative geometry. Take a line $L\subset\mathbb{P}^3$. The quadric surfaces in $\mathbb{P}^3$ that are tangent to $L$ are parametrized by a ...
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7 votes
0 answers
197 views

Geometric foundation of the Grothendieck polynomials

Grothendieck polynomials were firstly defined in Alain Lascoux and Marcel-Paul Sch¨utzenberger. Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une vari´et´e de drapeaux....
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3 votes
1 answer
149 views

Vanishing locus generic section $(\mathrm{sym}^2 \mathcal{R})(1)$

Let $n = 2m$ be an even integer and let $\mathcal{R}$ the tautological bundle on the Grassmannian $\mathrm{Gr}(2,n)$. I am looking for an explicit description of the degener The bundle $(\mathrm{Sym}^...
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4 votes
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What is a fast way to multiply a Schubert polynomial by an elementary symmetric polynomial (specifically $x_1\cdots x_k$)?

What is a computationally fast way to get the coefficients of Schubert polynomials in the expansion of the product of a Schubert polynomial and an elementary symmetric polynomial? I know "fast" is ...
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1 vote
1 answer
135 views

Proofs by Schubert calculus and combinatorics

Do you know some examples proved by two different methods: 1. Schubert calculus, 2. combinatorial method.
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Generically intersecting Schubert cycles

I have a question about the the proof of Pieri's formula from Harris' and Eisenbuds's lecture "3264 and All That"on page 146. Before the proof we use this terminology (see page 139): let $G=G(k,V)$...
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4 votes
1 answer
228 views

Transition equations for double Schubert polynomials

For $w$ a permutation, the associated (ordinary) Schubert polynomial $\mathfrak{S}_w(\textbf{x})$ is a multivariate polynomial that represents for the cohomology class of the Schubert variety $X_w$ in ...
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3 votes
1 answer
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Properties of a general element of the intersection of two Schubert cycles

We have the following lemma: Lemma Let $\Sigma_a(\mathcal{V}),\Sigma_b(\mathcal{W})$ be two Schubert cycles defined relative to transverse flags $\mathcal{V}$ and $\mathcal{W}$. If $\Lambda \in \...
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1 answer
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Schubert cycles that intersect generically transversely

Let $\mathcal{V}= 0 \subset V_1 \subset \cdots \subset V_{n-1}\subset V_n=V$, $\mathcal{W}=0 \subset W_1 \subset \cdots \subset W_{n-1} \subset W_n=W$ be two flags. We say that $\mathcal{V}$ and $\...
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0 answers
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Canonical sheaf of Schubert cycles

Suppose we have a smooth subvariety $X\subset Gr(2,n)$ of a Grassmannian, that can be expressed as usual as a linear combination of Schubert cycles. I would like to obtain information on the canonical ...
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Kac-Moody groups for non-crystallographic root systems

Given a finite-dimensional crystallographic root system, we can construct an associated Kac-Moody group, with a corresponding flag variety and Littlewood-Richardson coefficients. Between a pair of ...
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7 votes
1 answer
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Geometric interpretations of nil-Hecke ring and affine Hecke algebra

I am interested in two related constructions which give us either the cohomology or the $T \times \mathbb{C}^*$-equivariant $K$-theory of flag varieties. Let $G$ be a semisimple, simply connected ...
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2 votes
1 answer
162 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 ...
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8 votes
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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 $a < b$ ...
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10 votes
1 answer
455 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 ...
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1 vote
1 answer
181 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 ...
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5 votes
0 answers
151 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 ...
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3 votes
0 answers
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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(\...
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1 vote
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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_\...
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6 votes
2 answers
404 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-...
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5 votes
0 answers
276 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 ...
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11 votes
0 answers
219 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 ...
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230 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 ...
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1 vote
0 answers
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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 ...
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14 votes
3 answers
520 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,...
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5 votes
0 answers
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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 ...
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9 votes
1 answer
348 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 $\beta$. ...
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1 vote
1 answer
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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 ...
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5 votes
2 answers
241 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 ...
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6 votes
1 answer
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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)...
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14 votes
2 answers
965 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 ...
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5 votes
1 answer
568 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 ...
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4 votes
0 answers
151 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 ...
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3 votes
0 answers
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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 ...
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4 votes
1 answer
422 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\...
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1 vote
0 answers
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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 ...
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1 vote
0 answers
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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 ...
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10 votes
1 answer
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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}=...
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5 votes
0 answers
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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 ...
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3 votes
0 answers
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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 ...
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0 votes
1 answer
177 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 ...
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11 votes
0 answers
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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}...
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2 votes
0 answers
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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 ...
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5 votes
1 answer
121 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 ...
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  • 1,474
10 votes
1 answer
370 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 ...
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9 votes
1 answer
387 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 ...
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