# Questions tagged [schubert-calculus]

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79
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### Nonnegativity of the coefficients of the commuting difference operators of Fomin, Gelfand, and Postnikov evaluated on quantum Schubert polynomials

This post is about quantum Schubert polynomials. Fomin, Gelfand, and Postnikov defined operators in the nil-Hecke ring with coefficients in $\mathbb{Z}[x,q]$ denoted by $\chi_k$ for $1\leq k\leq n$ ...

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### Is this formula for certain structure constants of quantum Schubert polynomials known?

Quantum Schubert polynomials $\mathfrak{S}_u^q(x)$ indexed by $S_\infty$ are polynomials in the polynomial ring $\mathbb{Z}[x,q]$ in infinitely many variables that form a basis of this ring over $\...

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### Leibniz formula for Fulton's divided difference operators for quantum Schubert polynomials

Schubert polynomials are polynomials in the ring $\mathbb{Z}[x]$ where $x$ is an infinite set of variables indexed by the positive integers and they can be expressed in terms of "standard ...

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### Parabolic (double) quantum Schubert polynomials Pieri formula

I am writing calculation software for computing structure constants of equivariant quantum Schubert polynomials and I discovered that partial flag varieties corresponding to parabolic subgroups have ...

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### Expansion of Schubert polynomials into standard elementary monomials

I have an explicit formula for expressing any Schubert polynomial in terms of standard elementary monomials that may or may not be cancelation-free. I haven't determined this yet, but it seems likely ...

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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).$...

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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 ...

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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{...

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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 ...

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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,...

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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 ...

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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 \...

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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, ...

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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 ...

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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 $\...

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...