Questions tagged [schubert-calculus]

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Defining ideal of a Schubert variety as a kernel

Consider the Plücker embedding of the variety of complete flags in $\mathbb C^n$: $$F_n\subset\mathbb P(\bigwedge\nolimits^1\mathbb C^n)\times\dots\times\mathbb P(\bigwedge\nolimits^{n-1}\mathbb C^n).$...
Igor Makhlin's user avatar
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Necessary and/or sufficient conditions for LR coefficients of Schubert polynomials to be zero

There is a simple condition for determining whether LR coefficients for Schur polynomials are $0$, without invoking the Littlewood-Richardson rule. Is there anything similar, possibly weaker, for ...
Matt Samuel's user avatar
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"Sufficiently fat" Schur polynomial times Schubert polynomial is Schubert-positive

It's still open to find a combinatorial proof that for integers $k>0$ and $m>0$, a permutation $u\in S_\infty$ with its final descent at position $m$, and a partition $\lambda$ that $$\mathfrak{...
Matt Samuel's user avatar
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4 votes
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A basis for the 0-Hecke ring

Let $(W,S)$ be a Coxeter system of type $A_n$, with $$S=\{s_1,\ldots,s_n\}$$ satisfying the usual relations, and let $R=\mathbb{Z}[x_1,\ldots,x_{n+1}]$ be a polynomial ring. $W$ acts on $R$ by ...
Matt Samuel's user avatar
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5 votes
1 answer
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Pulling out a variable from a Schubert polynomial

Let $\mathfrak S_w(x_1,\ldots,x_n)$ be a Schubert polynomial. It's known that if we pick an index $i$, there are nonnegative integer coefficients $c_{w'}^w(i,j)$ such that $$\mathfrak S_w(x_1,\ldots,...
Matt Samuel's user avatar
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4 votes
1 answer
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Is this simple symmetry of Littlewood-Richardson coefficients known?

Let $\lambda$ be a partition with at most $p$ parts, let $\mu$ be a partition with at most $q$ parts, and let $\nu$ be a partition with at most $p+q$ parts. Let $m\geq \nu_1$ be an integer. We denote ...
Matt Samuel's user avatar
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6 votes
4 answers
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The dimension of the Grassmannian cohomology ring $H^*(\mathrm{Gr}_{n,d})$ and the fundamental $\frak{sl}_n$-representation $V_{\pi_d}$

The vector space dimension of the cohomology group of the $2$-plane Grassmannian $\mathrm{Gr}_{2,n}$ is given by the number of tuples $(\lambda_1,\lambda_2)$ satisfying $$ n - 2 \geq \lambda_1 \geq \...
Didier de Montblazon's user avatar
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Intersection of schubert varieties

Let $L_1$ and $L_2$ $\in$ $\mathbb{P}^4$ be two planes that intersect in exactly one point $Q$. Let $P_1 \in L_1$, $P_2 \in L_2$ points, such that $P_1 \neq Q \neq P_2$. Using the duality theorem, ...
user1131059's user avatar
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Schubert calculus and the representation ring of the general linear algebra

Schubert calculus studies the structural constants of the standard basis of the cohomology ring of the quantum Grassmannians. It is well known that it is isomorphic to the fusion ring of the category ...
Didier de Montblazon's user avatar
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Formulas for special elements of the nil-Hecke ring

Kostant and Kumar introduced the nil-Hecke ring for a crystallographic Coxeter group, which we will take to be $S_\infty$, which is the ring generated as a left module over the polynomial ring $\...
Matt Samuel's user avatar
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5 votes
1 answer
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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 ...
Blazej's user avatar
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1 answer
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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 ...
david_2020's user avatar
11 votes
1 answer
412 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 ...
Elsa's user avatar
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0 answers
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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 ...
FreddyG's user avatar
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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 ...
user avatar
7 votes
1 answer
439 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....
Cubic Bear's user avatar
3 votes
1 answer
192 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}^...
Libli's user avatar
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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 ...
Matt Samuel's user avatar
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1 vote
1 answer
143 views

Proofs by Schubert calculus and combinatorics

Do you know some examples proved by two different methods: 1. Schubert calculus, 2. combinatorial method.
Mihawk's user avatar
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0 answers
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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
319 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 ...
Zach H's user avatar
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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 \...
klerk's user avatar
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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 $\...
klerk's user avatar
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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 ...
IMeasy's user avatar
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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 ...
Matt Samuel's user avatar
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9 votes
1 answer
524 views

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 ...
Marc Besson's user avatar
2 votes
1 answer
190 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 ...
Sebastian K.'s user avatar
8 votes
0 answers
199 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 $a < b$ ...
Matt Samuel's user avatar
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10 votes
1 answer
511 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 ...
Pierre Dubois's user avatar
1 vote
1 answer
241 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 ...
user avatar
5 votes
0 answers
157 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 ...
Christoph Mark's user avatar
3 votes
0 answers
229 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(\...
Ben's user avatar
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0 answers
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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_\...
Ben's user avatar
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6 votes
2 answers
459 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-...
Rene Schipperus's user avatar
5 votes
0 answers
343 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 ...
Joseph O'Rourke's user avatar
12 votes
1 answer
377 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 ...
Christoph Mark's user avatar
0 votes
0 answers
244 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 ...
Lars Pettersen's user avatar
1 vote
0 answers
124 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 ...
Lars Pettersen's user avatar
14 votes
3 answers
579 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,...
Han Jin Ma's user avatar
5 votes
0 answers
147 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 ...
Han Jin Ma's user avatar
9 votes
1 answer
378 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 $\beta$. ...
Oliver's user avatar
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2 votes
1 answer
358 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 ...
MIQ's user avatar
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5 votes
2 answers
279 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 ...
Christoph Mark's user avatar
6 votes
1 answer
611 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)...
Lars Pettersen's user avatar
14 votes
2 answers
1k 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 ...
Moritz Firsching's user avatar
5 votes
1 answer
688 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 ...
Pavel Katzo's user avatar
4 votes
0 answers
159 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 ...
Jake Levinson's user avatar
3 votes
0 answers
182 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 ...
Lionel's user avatar
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5 votes
1 answer
446 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\...
Lionel's user avatar
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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 ...
Christoph Mark's user avatar