arXiv subjects

Allen Knutson

Publications and source records attributed to Allen Knutson.

At least 37 records · Page 2Linked to original sources

arXiv2014

Positive formulae for K-types of SL_3(R)-irreps and a Blattner formula for smooth K-orbit closures

We prove a version of Blattner's conjecture, for irreducible subquotients of principal series representations with integral infinitesimal character of a real reductive Lie group whose Beilinson-Bernstein D-module is supported on a K-orbit with smooth closure. (The cases usually considered are closed orbits, or their preimages along G/B -> G/P.) We apply this to G_R = SL_3(R), where all four K-orbits on G/B have smooth closure, and refine the resulting alternating-sum formulae to ones with only positive terms.

math.RT↗

arXiv2014

The Brauer loop scheme and orbital varieties

A. Joseph invented multidegrees in [Jo84] to study orbital varieties, which are the components of an orbital scheme, itself constructed by intersecting a nilpotent orbit with a Borel subalgebra. Their multidegrees, known as Joseph polynomials, give a basis of a (Springer) representation of the Weyl group. In the case of the nilpotent orbit $\{ M^2=0 \}$, the orbital varieties can be indexed by noncrossing chord diagrams in the disc. In this paper we study the normal cone to the orbital scheme inside this nilpotent orbit $\{ M^2 = 0 \}$. This gives a better-motivated construction of the Brauer loop scheme we introduced in [KZJ07], whose components are indexed by all chord diagrams (now possibly with crossings) in the disc. The multidegrees of its components, the Brauer loop varieties, were shown to reproduce the ground state of the Brauer loop model in statistical mechanics [DFZJ06]. Here, we reformulate and slightly generalize these multidegrees in order to express them as solutions of the rational quantum Knizhnik--Zamolodchikov equation associated to the Brauer algebra. In particular, the vector of the multidegrees satisfies two sets of equations, corresponding to the $e_i$ and $f_i$ generators of the Brauer algebra. We describe here the geometric meaning of both $e_i$ and $f_i$ equations in our slightly extended setting. We also describe the corresponding actions at the level of orbital varieties: while only the $e_i$ equations make sense directly on the Joseph polynomials, the $f_i$ equations also appear if one introduces a broader class of varieties. We explain the connection of the latter with matrix Schubert varieties.

math.AG↗

arXiv2014

Schubert calculus and shifting of interval positroid varieties

Consider k x n matrices with rank conditions placed on intervals of columns. The ranks that are actually achievable correspond naturally to upper triangular partial permutation matrices, and we call the corresponding subvarieties of Gr(k,n) the _interval positroid varieties_, as this class lies within the class of positroid varieties studied in [Knutson-Lam-Speyer]. It includes Schubert and opposite Schubert varieties, and their intersections, and is Grassmann dual to the projection varieties of [Billey-Coskun]. Vakil's "geometric Littlewood-Richardson rule" [Vakil] uses certain degenerations to positively compute the H^*-classes of Richardson varieties, each summand recorded as a (2+1)-dimensional "checker game". We use his same degenerations to positively compute the K_T-classes of interval positroid varieties, each summand recorded more succinctly as a 2-dimensional "K-IP pipe dream". In Vakil's restricted situation these IP pipe dreams biject very simply to the puzzles of [Knutson-Tao]. We relate Vakil's degenerations to Erd\H os-Ko-Rado shifting, and include results about computing "geometric shifts" of general T-invariant subvarieties of Grassmannians.

math.AG↗

arXiv2012

Singularities of Richardson varieties

We give a short proof that essentially all questions concerning singularities of Richardson varieties reduce to corresponding questions about Schubert varieties. Consequently, we quickly deduce some new and previously known results.

math.AG↗

arXiv2011

Positroid Varieties: Juggling and Geometry

While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, the intersection of only the cyclic shifts of one Bruhat decomposition turns out to have many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown-Goodearl-Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from the authors' previous work, we show that positroid varieties are normal, Cohen-Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plucker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This last fact lets us connect Postnikov's and Buch-Kresch-Tamvakis' approaches to quantum Schubert calculus.

math.AG↗

arXiv2011

Projections of Richardson Varieties

While the projections of Schubert varieties in a full generalized flag manifold G/B to a partial flag manifold $G/P$ are again Schubert varieties, the projections of Richardson varieties (intersections of Schubert varieties with opposite Schubert varieties) are not always Richardson varieties. The stratification of G/P by projections of Richardson varieties arises in the theory of total positivity and also from Poisson and noncommutative geometry. In this paper we show that many of the geometric properties of Richardson varieties hold more generally for projected Richardson varieties; they are normal, Cohen-Macaulay, have rational singularities, and are compatibly Frobenius split with respect to the standard splitting. Indeed, we show that the projected Richardson varieties are the only compatibly split subvarieties, providing an example of the recent theorem [Schwede, Kumar-Mehta] that a Frobenius split scheme has only finitely many compatibly split subvarieties. (The G/B case was treated by [Hague], whose proof we simplify somewhat.) One combinatorial analogue of a Richardson variety is the order complex of the corresponding Bruhat interval in W; this complex is known to be an EL-shellable ball [Bjorner-Wachs '82]. We prove that the projection of such a complex into the order complex of the Bruhat order on W/W_P is again a shellable ball. This requires extensive analysis of "P-Bruhat order", a generalization of the k-Bruhat order of [Bergeron-Sottile '98]. In the case that G/P is minuscule (e.g. a Grassmannian), we show that its Grobner degeneration takes each projected Richardson variety to the Stanley-Reisner scheme of its corresponding ball.

math.AG↗

arXiv2010

Product and puzzle formulae for GL_n Belkale-Kumar coefficients

The Belkale-Kumar product on H*(G/P) is a degeneration of the usual cup product on the cohomology ring of a generalized flag manifold. In the case G=GL_n, it was used by N. Ressayre to determine the regular faces of the Littlewood-Richardson cone. We show that for G/P a (d-1)-step flag manifold, each Belkale-Kumar structure constant is a product of d(d-1)/2 Littlewood-Richardson numbers, for which there are many formulae available, e.g. the puzzles of [Knutson-Tao '03]. This refines previously known factorizations into d-1 factors. We define a new family of puzzles to assemble these to give a direct combinatorial formula for Belkale-Kumar structure constants. These "BK-puzzles" are related to extremal honeycombs, as in [Knutson-Tao-Woodward~'04]; using this relation we give another proof of Ressayre's result. Finally, we describe the regular faces of the Littlewood-Richardson cone on which the Littlewood-Richardson number is always 1; they correspond to nonzero Belkale-Kumar coefficients on partial flag manifolds where every subquotient has dimension 1 or 2.

math.CO↗

arXiv2010

Puzzles, positroid varieties, and equivariant K-theory of Grassmannians

Vakil studied the intersection theory of Schubert varieties in the Grassmannian in a very direct way: he degenerated the intersection of a Schubert variety X_mu and opposite Schubert variety X^nu to a union {X^lambda}, with repetition. This degeneration proceeds in stages, and along the way he met a collection of more complicated subvarieties, which he identified as the closures of certain locally closed sets. We show that Vakil's varieties are _positroid varieties_, which in particular shows they are normal, Cohen-Macaulay, have rational singularities, and are defined by the vanishing of Plücker coordinates [Knutson-Lam-Speyer]. We determine the equations of the Vakil variety associated to a partially filled ``puzzle'' (building on the appendix to [Vakil]), and extend Vakil's proof to give a geometric proof of the puzzle rule from [Knutson-Tao '03] for equivariant Schubert calculus. The recent paper [Anderson-Griffeth-Miller] establishes (abstractly; without a formula) three positivity results in equivariant K-theory of flag manifolds G/P. We demonstrate one of these concretely, giving a corresponding puzzle rule.

math.AG↗

arXiv2009

Frobenius splitting, point-counting, and degeneration

Let f be a polynomial of degree n in ZZ[x_1,..,x_n], typically reducible but squarefree. From the hypersurface {f=0} one may construct a number of other subschemes {Y} by extracting prime components, taking intersections, taking unions, and iterating this procedure. We prove that if the number of solutions to f=0 in \FF_p^n is not a multiple of p, then all these intersections in Å^n_{\FF_p} just described are reduced. (If this holds for infinitely many p, then it holds over \QQ as well.) More specifically, there is a_Frobenius splitting_ on Å^n_{\FF_p} compatibly splitting all these subschemes {Y}. We determine when a Gröbner degeneration f_0=0 of such a hypersurface f=0 is again such a hypersurface. Under this condition, we prove that compatibly split subschemes degenerate to compatibly split subschemes, and stay reduced. Our results are strongest in the case that f's lexicographically first term is \prod_{i=1}^n x_i. Then for all large p, there is a Frobenius splitting that compatibly splits f's hypersurface and all the associated {Y}. The Gröbner degeneration Y' of each such Y is a reduced union of coordinate spaces (a Stanley-Reisner scheme), and we give a result to help compute its Gröbner basis. We exhibit an f whose associated {Y} include Fulton's matrix Schubert varieties, and recover much more easily the Gröbner basis theorem of [Knutson-Miller '05]. We show that in Bott-Samelson coordinates on an opposite Bruhat cell X^v_\circ in G/B, the f defining the complement of the big cell also has initial term \prod_{i=1}^n x_i, and hence the Kazhdan-Lusztig subvarieties {X^v_{w\circ}} degenerate to Stanley-Reisner schemes. This recovers, in a weak form, the main result of [Knutson '08].

math.AG↗

arXiv2009

Positroid varieties I: juggling and geometry

While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, the intersection of only the {\em cyclic shifts} of one Bruhat decomposition turns out to have many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, and Brown-Goodearl-Yakimov. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call {\em bounded juggling patterns}. We adopt his terminology and call the strata {\em positroid varieties.} We show that positroid varieties are normal and Cohen-Macaulay, and are defined as schemes by the vanishing of Plucker coordinates. We compute their T-equivariant Hilbert series, and show that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov's and Buch-Kresch-Tamvakis' approaches to quantum Schubert calculus. Our principal tools are the Frobenius splitting results for Richardson varieties as developed by Brion, Lakshmibai, and Littelmann, and the Hodge-Grobner degeneration of the Grassmannian. We show that each positroid variety degenerates to the projective Stanley-Reisner scheme of a shellable ball.

math.AG↗

arXiv2009

Frobenius splitting and Möbius inversion

We show that the fundamental class in K-homology of a Frobenius split scheme can be computed as a certain alternating sum over irreducible varieties, with the coefficients computed using Möbius inversion on a certain poset. If G/P is a generalized flag manifold and X is an irreducible subvariety homologous to a multiplicity-free union of Schubert varieties, then using a result of Brion we show how to compute the K_0-class [X] in K_0(G/P) from the Chow class in A_*(G/P).

math.AG↗

arXiv2009

Compatibly Frobenius split subschemes are rigid

Schwede proved very recently in arXiv:0901.1154 that in a quasiprojective scheme X with a fixed Frobenius splitting, there are only finitely many subschemes {Y} that are compatibly split. (A simpler proof has already since been given in arXiv:0901.2098, by Kumar and Mehta.) It follows that their deformations (as compatibly split subschemes) are obstructed. We give a short proof that if X is projective, its compatibly split subschemes {Y} have no deformations at all (again, as compatibly split subschemes). This reproves Schwede's result in some simple cases.

math.AG↗

arXiv2008

Schubert patches degenerate to subword complexes

We study the intersections of general Schubert varieties X_w with permuted big cells, and give an inductive degeneration of each such "Schubert patch" to a Stanley-Reisner scheme. Similar results had been known for Schubert patches in various types of Grassmannians. We maintain reducedness using the results of [Knutson 2007] on automatically reduced degenerations, or through more standard cohomology-vanishing arguments. The underlying simplicial complex of the Stanley-Reisner scheme is a subword complex, as introduced for slightly different purposes in [Knutson-Miller 2004], and is homeomorphic to a ball. This gives a new proof of the Andersen-Jantzen-Soergel/Billey and Graham/Willems formulae for restrictions of equivariant Schubert classes to fixed points.

math.AG↗

arXiv2008

A compactly supported formula for equivariant localization, and, simplicial complexes of Bialynicki-Birula decompositions

Let X be a projective scheme carrying a circle action S with isolated fixed points. We associate a simplicial complex Delta(X,S) of "closure chains" using a refinement of its Morse/Bialynicki-Birula decomposition. If this decomposition is a stratification (e.g. when X is a flag manifold), then Delta(X,S) is just the order complex of the poset of fixed points. For X a toric variety, Delta(X,S) is a triangulation of the moment polytope. We compute some other examples, including a Bott-Samelson manifold and the punctual Hilbert scheme of 4 points in the plane. Summing over the facets of Delta(X,S), we obtain a positive formula for the Duistermaat-Heckman measure on the moment polytope of X, defined for any torus action extending S. We explain how, through brutal use of partial fractions, this can be extended to an AB/BV-type formula for integrating general classes. Throughout we work with equivariant Chow groups, and do not make any smoothness requirements on X.

math.AG↗

arXiv2007

Automatically reduced degenerations of automatically normal varieties

Let F be a flat family of projective schemes, whose geometric generic fiber is reduced and irreducible. We give conditions on a special fiber (a "limit" of the family) to guarantee that it too is reduced. These conditions often imply also that the generic fiber is normal. The conditions are particularly easy to check in the setup of a "geometric vertex decomposition" [Knutson-Miller-Yong '07]. The primary tool used is the corresponding limit _branchvariety_ [Alexeev-Knutson '06], which is reduced by construction, and maps to the limit subscheme; our technique is to use normality to show that the branchvariety map must be an isomorphism. As a demonstration, we give an essentially naive proof that Schubert varieties in finite type are normal and Cohen-Macaulay. The proof does not involve any resolution of singularities or cohomology-vanishing techniques (e.g. appeal to characteristic p).

math.AG↗

arXiv2007

Orbifold cohomology of torus quotients

We introduce the_inertial cohomology ring_ NH^*_T(Y) of a stably almost complex manifold carrying an action of a torus T. We show that in the case that Y has a locally free action by T, the inertial cohomology ring is isomorphic to the Chen-Ruan orbifold cohomology ring H_{CR}^*(Y/T) of the quotient orbifold Y/T. For Y a compact Hamiltonian T-space, we extend to orbifold cohomology two techniques that are standard in ordinary cohomology. We show that NH^*_T(Y) has a natural ring surjection onto H_{CR}^*(Y//T), where Y//T is the symplectic reduction of Y by T at a regular value of the moment map. We extend to NH^*_T(Y) the graphical GKM calculus (as detailed in e.g. [Harada-Henriques-Holm]), and the kernel computations of [Tolman-Weitsman, Goldin]. We detail this technology in two examples: toric orbifolds and weight varieties, which are symplectic reductions of flag manifolds. The Chen-Ruan ring has been computed for toric orbifolds, with \Q coefficients, in [Borisov-Chen-Smith]); symplectic toric orbifolds obtained by reduction by a connected torus (though with different computational methods), and extend them to \Z coefficients in certain cases, including weighted projective spaces.

math.SG↗

arXiv2006

A scheme related to the Brauer loop model

We introduce the_Brauer loop scheme_ E := {M in M_N(C) : M\cp M = 0}, where \cp is a certain degeneration of the ordinary matrix product. Its components of top dimension, floor(N^2/2), correspond to involutions πin S_N having one or no fixed points. In the case N even, this scheme contains the upper-upper scheme from [Knutson '04] as a union of (N/2)! of its components. One of those is a degeneration of the_commuting variety_ of pairs of commuting matrices. The_Brauer loop model_ is a quantum integrable stochastic process introduced in [de Gier--Nienhuis '04], and some of the entries of its Perron-Frobenius eigenvector were observed (conjecturally) to match the degrees of the components of the upper-upper scheme. We extend this, with proof, to_all_ the entries: they are the degrees of the components of the Brauer loop scheme. Our proof of this follows the program outlined in [Di Francesco--Zinn-Justin '04]. In that paper, the entries of the Perron-Frobenius eigenvector were generalized from numbers to polynomials, which allowed them to be calculated inductively using divided difference operators. We relate these polynomials to the multidegrees of the components of the Brauer loop scheme, defined using an evident torus action on E. In particular, we obtain a formula for the degree of the commuting variety, previously calculated up to 4x4 matrices.

math.AG↗

arXiv2006

Kempf collapsing and quiver loci

Kempf [1976] studied proper, G-equivariant maps from equivariant vector bundles over flag manifolds to G-representations V, which he called _collapsings_. We give a simple formula for the G-equivariant cohomology class on V, or_multidegree_, associated to the image of a collapsing: apply a certain sequence of divided difference operators to a certain product of linear polynomials, then divide by the number of components in a general fiber. When that number of components is 1, we construct a desingularization of the image of the collapsing. If in addition the image has rational singularities, we can use the desingularization to give also a formula for the G-equivariant K-class of the image, whose leading term is the multidegree. Our application is to quiver loci and quiver polynomials. Let Q be a quiver of finite type (A, D, or E, in arbitrary orientation), and assign a vector space to each vertex. Let \Hom denote the (linear) space of representations of Q with these vector spaces. This carries an action of GL, the product of the general linear groups of the individual vector spaces. A_quiver locus_ Ωis the closure in \Hom of a GL-orbit, and its multidegree is the corresponding _quiver polynomial_. Reineke [2004] proved that every ADE quiver locus is the image of a birational Kempf collapsing (giving a desingularization directly). Using Reineke's collapsings, we give formulae for ADE quiver polynomials, previously only computed in type A (though in this case, our formulae are new). In the A and D cases quiver loci are known to have rational singularities [Bobiński-Zwara 2002], so we also get formulae for their K-classes, which had previously only been computed in equioriented type A (and again our formulae are new).

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