Minimal Schubert Varieties Admitting Semistable Points for Exceptional Cases

For any simple, simply connected algebraic group G of exceptional types (E ₆, E ₇, E ₈, F ₄, and G ₂) and for any maximal parabolic subgroup P of G, we describe all minimal (with respect to inclusion) Schubert varieties in G/P admitting semistable points for the action of a maximal torus T with respect to an ample line bundle on G/P. This completes the answer to a question proposed in [8 Kannan , S. S. , Pattanayak , S. K. ( 2009 ). Torus quotients of homogeneous spaces: Minimal dimensional Schubert varieties admitting semi-stable points . Proc. Indian Acad. Sci. (Math. Sci.) 119 ( 4 ): 469 – 485 .[Crossref], [Web of Science ®] , [Google Scholar]] and settled there in the classical case.     

Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor

The intersection ring of a complex Grassmann manifold is generated by Schubert varieties, and its structure is governed by the Littlewood-Richardson rule. Given three Schubert varieties S₁, S₂, S₃ with intersection number equal to one, we show how to construct an explicit element in their intersection. This element is obtained generically as the result of a sequence of lattice operations on the spaces of the corresponding flags, and is therefore well defined over an arbitrary field of scalars. Moreover, this result also applies to appropriately defined analogues of Schubert varieties in the Grassmann manifolds associated with a finite von Neumann algebra. The arguments require the combinatorial structure of honeycombs, particularly the structure of the rigid extremal honeycombs. It is known that the eigenvalue distributions of self-adjoint elements a,b,c with a+b+c=0 in the factor Rω are characterized by a system of inequalities analogous to the classical Horn inequalities of linear algebra. We prove that these inequalities are in fact true for elements of an arbitrary finite factor. In particular, if x,y,z are self-adjoint elements of such a factor and x+y+z=0, then there exist self-adjoint a,b,c∈Rω such that a+b+c=0 and a (respectively, b,c) has the same eigenvalue distribution as x (respectively, y,z). A (‘complete’) matricial form of this result is known to imply an affirmative answer to an embedding question formulated by Connes. The critical point in the proof of this result is the production of elements in the intersection of three Schubert varieties. When the factor under consideration is the algebra of n×n complex matrices, our arguments provide new and elementary proofs of the Horn inequalities, which do not require knowledge of the structure of the cohomology of the Grassmann manifolds.     

Initial ideals of tangent cones to Schubert varieties in orthogonal Grassmannians

We compute the initial ideals, with respect to certain conveniently chosen term orders, of ideals of tangent cones at torus fixed points to Schubert varieties in orthogonal Grassmannians. The initial ideals turn out to be square-free monomial ideals and therefore define Stanley-Reisner face rings of simplicial complexes. We describe these complexes. The maximal faces of these complexes encode certain sets of non-intersecting lattice paths.     

Schubert varieties, linear codes and enumerative combinatorics

We consider linear error correcting codes associated to higher-dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult questions in combinatorics and algebraic geometry. This is illustrated by codes associated to Schubert varieties in Grassmannians, called Schubert codes, which have recently been studied. The basic parameters such as the length, dimension and minimum distance of these codes are known only in special cases. An upper bound for the minimum distance is known and it is conjectured that this bound is achieved. We give explicit formulae for the length and dimension of arbitrary Schubert codes and prove the minimum distance conjecture in the affirmative for codes associated to Schubert divisors.     

Bioriented flags and resolutions of Schubert varieties

We use incidence relations running in two directions in order to construct a Kempf–Laksov type resolution for any Schubert variety of the complete flag manifold but also an embedded resolution for any Schubert variety in the Grassmannian. These constructions are alternatives to the celebrated Bott–Samelson resolutions. The second process led to the introduction of W-flag varieties, algebro-geometric objects that interpolate between the standard flag manifolds and products of Grassmannians, but which are singular in general. The surprising simple desingularization of a particular such type of variety produces an embedded resolution of the Schubert variety within the Grassmannian.     

Flexibility of Schubert classes

In this note, we discuss the flexibility of Schubert classes in homogeneous varieties. We give several constructions for representing multiples of a Schubert class by irreducible subvarieties. We sharpen [22, Theorem 3.1] by proving that every positive multiple of an obstructed class in a cominuscule homogeneous variety can be represented by an irreducible subvariety.     

Cohomology of line bundles on Schubert varieties: The rank two case

In this paper we describe vanishing and non-vanishing of cohomology of “most” line bundles over Schubert subvarieties of flag varieties for rank 2 semisimple algebraic groups.     

The Gorenstein locus of minuscule Schubert varieties

In this article, we explicitly describe the Gorenstein locus of all minuscule Schubert varieties. This proves a special case of the conjecture of A. Woo and A. Yong [Alexander Woo, Alexander Yong, Governing singularities of Schubert varieties, J. Algebra 320 (2) (2008) 495-520] regarding the Gorenstein locus of Schubert varieties.     

Pairs of Semisimple Lie Algebras and their Maximal Reductive Subalgebras

Let \(\mathfrak g\) be a semisimple Lie algebra over a field \(\mathbb K\), \(\text{char}\left( \mathbb{K} \right)=0\), and \(\mathfrak g_1\) a subalgebra reductive in \(\mathfrak g\). Suppose that the restriction of the Killing form B of \(\mathfrak g\) to \(\mathfrak g_1 \times \mathfrak g_1\) is nondegenerate. Consider the following statements: ( 1) For any Cartan subalgebra \(\mathfrak h_1\) of \(\mathfrak g_1\) there is a unique Cartan subalgebra \(\mathfrak h\) of \(\mathfrak g\) containing \(\mathfrak h_1\); ( 2) \(\mathfrak g_1\) is self-normalizing in \(\mathfrak g\); ( 3) The B-orthogonal \(\mathfrak p\) of \(\mathfrak g_1\) in \(\mathfrak g\) is simple as a \(\mathfrak g_1\)-module for the adjoint representation. We give some answers to this natural question: For which pairs \((\mathfrak g,\mathfrak g_1)\) do ( 1), ( 2) or ( 3) hold? We also study how \(\mathfrak p\) in general decomposes as a \(\mathfrak g_1\)-module, and when \(\mathfrak g_1\) is a maximal subalgebra of \(\mathfrak g\). In particular suppose \((\mathfrak g,\sigma )\) is a pair with \(\mathfrak g\) as above and σ its automorphism of order m. Assume that \(\mathbb K\) contains a primitive m-th root of unity. Define \(\mathfrak g_1:=\mathfrak g^{\sigma}\), the fixed point algebra for σ. We prove the following generalization of a well known result for symmetric Lie algebras, i.e., for m=2: (a) \((\mathfrak g,\mathfrak g_1)\) satisfies ( 1); (b) For m prime, \((\mathfrak g,\mathfrak g_1)\) satisfies ( 2).     

Enumerating indices of Schubert varieties defined by inclusions

By extending the notion of grid classes to include infinite grids, we establish a structural characterisation of the simple permutations in Av(4231,35142,42513,351624)$\mathrm{Av}(4231,35142,42513,351624)$, a pattern class which has three different connections with algebraic geometry, including the specification of indices of Schubert varieties defined by inclusions. This characterisation leads to the enumeration of the class.     

Twisted loop groups and their affine flag varieties

We develop a theory of affine flag varieties and of their Schubert varieties for reductive groups over a Laurent power series local field k((t)) with k a perfect field. This can be viewed as a generalization of the theory of affine flag varieties for loop groups to a “twisted case”; a consequence of our results is that our construction also includes the flag varieties for Kac–Moody Lie algebras of affine type. We also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k((t)) and some applications to local models of Shimura varieties.     

Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence

The Richardson variety X _α ^γ in the Grassmannian is defined to be the intersection of the Schubert variety X ^γ and opposite Schubert variety X _α . We give an explicit Gröbner basis for the ideal of the tangent cone at any T-fixed point of X _α ^γ , thus generalizing a result of Kodiyalam-Raghavan (J. Algebra 270(1):28–54, 2003) and Kreiman-Lakshmibai (Algebra, Arithmetic and Geometry with Applications, 2004). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the bounded RSK (BRSK). We use the Gröbner basis result to deduce a formula which computes the multiplicity of X _α ^γ at any T-fixed point by counting families of nonintersecting lattice paths, thus generalizing a result first proved by Krattenthaler (Sém. Lothar. Comb. 45:B45c, 2000/2001; J. Algebr. Comb. 22:273–288, 2005).     

Rigidity of smooth Schubert varieties in Hermitian symmetric spaces

In this paper we study the space of effective -cycles in with the homology class equal to an integral multiple of the homology class of Schubert variety of type . When is a proper linear subspace of a linear space in , we know that is already complicated. We will show that for a smooth Schubert variety in a Hermitian symmetric space, any irreducible subvariety with the homology class , , is again a Schubert variety of type , unless is a non-maximal linear space. In particular, any local deformation of such a smooth Schubert variety in Hermitian symmetric space is obtained by the action of the Lie group .

     

A strong constructive version of engel's theorem

We examine two problems in the computational theory of Lie algebras. First, we prove a constructive version of Engel's theorem: if L is a finite-dimensional Lie algebra that is not nilpotent, we show how to construct an element x in L such that the linear transformation ad x is not nilpotent. No special assumptions about the underlying field are needed. Second, as an important application of the first result, we give an algorithm for the construction of a Cartan subalgebra of a finite-dimensional Lie algebra. This solves the problem of finding a totally constructive proof of the existence of a Cartan subalgebra, posed by Beck, Kolman, and Stewart in the paper "Computing the Structure of a Lie Algebra". Our proofs are ordinary mathematical proofs that do not employ the general law of excluded middle. The advantage of this approach to mathematics is that our proofs, which are not burdened or obscured by the details of a particular programming language, can nevertheless be routinely turned into computer programs     

Desingularizations of Schubert varieties in double Grassmannians

Let X = Gr(k, V) × Gr(l, V) be the direct product of two Grassmann varieties of k-and l-planes in a finite-dimensional vector space V, and let B ? GL(V) be the isotropy group of a complete flag in V. We consider B-orbits in X, which are an analog of Schubert cells in Grassmannians. We describe this set of orbits combinatorially and construct desingularizations for the closures of these orbits, similar to the Bott-Samelson desingularizations for Schubert varieties.