# Questions tagged [schubert-calculus]

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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 $$n - 2 \geq \lambda_1 \geq \... 1 vote 0 answers 80 views ### Intersection of schubert varieties Let L_1 and L_2 \in \mathbb{P}^4 be two planes that intersect in exactly one point Q. Let P_1 \in L_1, P_2 \in L_2 points, such that P_1 \neq Q \neq P_2. Using the duality theorem, ... 1 vote 0 answers 112 views ### Schubert calculus and the representation ring of the general linear algebra Schubert calculus studies the structural constants of the standard basis of the cohomology ring of the quantum Grassmannians. It is well known that it is isomorphic to the fusion ring of the category ... 2 votes 0 answers 62 views ### Formulas for special elements of the nil-Hecke ring Kostant and Kumar introduced the nil-Hecke ring for a crystallographic Coxeter group, which we will take to be S_\infty, which is the ring generated as a left module over the polynomial ring \... 5 votes 1 answer 274 views ### Intersection cycle in a product of Grassmannians Let G(k,n) denote the Grassmiannian of k-planes in \mathbb C^n. Let's define$$ I_j =\{ (\Lambda_1,\Lambda_2 ) \in G(k,n) \times G(l,n) \, | \, \dim(\Lambda_1 \cap \Lambda_2) \geq j \}. $$These ... 5 votes 1 answer 436 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 ... 12 votes 1 answer 399 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 ... 0 votes 0 answers 98 views ### 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 ... 2 votes 0 answers 173 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 ... 7 votes 1 answer 375 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.... 3 votes 1 answer 174 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}^... 4 votes 0 answers 111 views ### 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 ... 1 vote 1 answer 140 views ### Proofs by Schubert calculus and combinatorics Do you know some examples proved by two different methods: 1. Schubert calculus, 2. combinatorial method. 1 vote 0 answers 71 views ### 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)... 4 votes 1 answer 263 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 ... 3 votes 1 answer 339 views ### 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 \... 1 vote 1 answer 222 views ### 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 \... 3 votes 0 answers 81 views ### 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 ... 5 votes 0 answers 84 views ### 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 ... 7 votes 1 answer 447 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 ... 2 votes 1 answer 178 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 0 answers 194 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 1 answer 485 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 1 answer 195 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 0 answers 154 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\,, $$where S_W is the coinvariant algebra of the Weyl ... 3 votes 0 answers 223 views ### T-equivariant homology of affine Grassmannian Let G=SL_n, denote the affine Grassmannian Gr:=Gr_{G}=\mathcal{G}/\mathcal{P}, where \mathcal{G}=G(\mathbb{C}((z))) and \mathcal{P}=G(\mathbb{C}[[z]]). We know that R:=H_*^T(Gr)\cong H_*^T(\... 1 vote 0 answers 120 views ### intersect a subvariety with a Schubert variety Let Y be an irreducible subvariety inside Gr(r,n) (Grassmannian of r-plane inside \mathbb{C}^n) and X_\lambda be a Schubert variety corresponding to \lambda. Assume that codim(Y)+codim(X_\... 6 votes 2 answers 435 views ### Some Elementary Schubert Calculus Calculations Here are some simple geometry problems I am unable to resolve to my satisfaction. I asked the question on Math Stack (https://math.stackexchange.com/questions/2713754/a-problem-in-elementary-... 5 votes 0 answers 317 views ### Number of bitangents to connected algebraic curve Schubert showed that a plane algebraic curve of degree d has at most$$ \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 0 answers 225 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 0 answers 237 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 0 answers 123 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 3 answers 547 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 0 answers 144 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 ... 9 votes 1 answer 358 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 1 answer 336 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 2 answers 253 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 1 answer 554 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)... 14 votes 2 answers 989 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 ... 5 votes 1 answer 608 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 0 answers 156 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 0 answers 179 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 ... 5 votes 1 answer 431 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 0 answers 140 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 0 answers 87 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 1 answer 180 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 0 answers 206 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 0 answers 58 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 ...
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 ...
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}...