arXiv subjects

Allen Knutson

Publications and source records attributed to Allen Knutson.

At least 55 records · Page 3Linked to original sources

arXiv2006

Complete moduli spaces of branchvarieties

The space of subvarieties of P^n with a fixed Hilbert polynomial is not complete. Grothendieck defined a completion by relaxing "variety" to "scheme", giving the complete_Hilbert scheme_ of subschemes of P^n with fixed Hilbert polynomial. We instead relax "sub" to "branch", where a_branchvariety of_ P^n is defined to be a_reduced_ (though possibly reducible) scheme_with a finite morphism to_ P^n. Our main theorems are that the moduli stack of branchvarieties of P^n with fixed Hilbert polynomial and total degrees of i-dimensional components is a proper (complete and separated) Artin stack with finite stabilizer, and has a coarse moduli space which is a proper algebraic space. Families of branchvarieties have many more locally constant invariants than families of subschemes; for example, the number of connected components is a new invariant. In characteristic 0, one can extend this count to associate a Z-labeled rooted forest to any branchvariety.

math.AG↗

arXiv2006

Tableau complexes

Let X,Y be finite sets and T a set of functions from X -> Y which we will call "tableaux". We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such "tableau complexes" have many nice properties, and are frequently homeomorphic to balls, which we prove using vertex decompositions. In our motivating example, the facets are labeled by semistandard Young tableaux, and the more general interior faces are labeled by Buch's set-valued semistandard tableaux. One vertex decomposition of this "Young tableau complex" parallels Lascoux's transition formula for vexillary double Grothendieck polynomials. Consequently, we obtain formulae (both old and new) for these polynomials. In particular, we present a common generalization of the formulae of Wachs and Buch, each of which implies the classical tableau formula for Schur polynomials.

math.CO↗

arXiv2005

Balanced normal cones and Fulton-MacPherson's intersection theory

Let X be a subscheme of a reduced scheme Y. Then Y has a flat "degeneration to the normal cone" C_X Y of X, and this degeneration plays a key step in Fulton and MacPherson's "basic construction" in intersection theory. The intersection product has a canonical refinement as a sum over the components of C_X Y, for X and Y depending on the given intersection problem. The cone C_X Y is usually not reduced, which leads to the appearance of multiplicities in intersection formulae. We describe a variant of this degeneration, due essentially to Samuel, Rees, and Nagata, in which Y flatly degenerates to the "balanced" normal cone \barC_X Y. This space is reduced, and has a natural map onto the reduction (C_X Y)_red of C_X Y. The multiplicity of a component now appears as the degree of this map. Hence intersection theory can be studied using only reduced schemes. Moreover, since the map \barC_X Y \to (C_X Y)_red may wrap multiple components of \barC_X Y around one component of C_X Y, writing the intersection product as a sum over the components of \barC_X Y gives a further canonical refinement. \\ In the case that X is a Cartier divisor in a projective scheme Y, we describe the balanced normal cone in homotopy-theoretic terms, and prove a useful upper bound on the Hilbert function of \barC_X Y.

math.AG↗

arXiv2005

Gröbner geometry of vertex decompositions and of flagged tableaux

We relate a classic algebro-geometric degeneration technique, dating at least to [Hodge 1941], to the notion of vertex decompositions of simplicial complexes. The good case is when the degeneration is reduced, and we call this a "geometric vertex decomposition". Our main example in this paper is the family of vexillary matrix Schubert varieties, whose ideals are also known as (one-sided) ladder determinantal ideals. Using a diagonal term order to specify the (Gröbner) degeneration, we show that these have geometric vertex decompositions into simpler varieties of the same type. From this, together with the combinatorics of the pipe dreams of [Fomin--Kirillov 1996], we derive a new formula for the numerators of their multigraded Hilbert series, the double Grothendieck polynomials, in terms of "flagged set-valued tableaux". This unifies work of [Wachs 1985] on flagged tableaux, and [Buch 2002] on set-valued tableaux, giving geometric meaning to both. This work focuses on diagonal term orders, giving results complementary to those of [Knutson--Miller 2004], where it was shown that the generating minors form a Gröbner basis for any antidiagonal term order and any matrix Schubert variety. We show here that under a diagonal term order, the only matrix Schubert varieties for which these minors form Gröbner bases are the vexillary ones, reaching an end toward which the ladder determinantal literature had been building.

math.AG↗

arXiv2004

A formula for K-theory truncation Schubert calculus

Define a ``truncation'' $r_{t}(p)$ of a polynomial $p$ in $\{x_1,x_2,x_3,...\}$ as the polynomial with all but the first $t$ variables set to zero. In certain good cases, the truncation of a Schubert or Grothendieck polynomial may again be a Schubert or Grothendieck polynomial. We use this phenomenon to give subtraction-free formulae for certain Schubert structure constants in $K(Flags({\mathbb C}^n))$, in particular generalizing those from [Kogan '00] in which only cohomology was treated, and from [Buch `02] on the Grassmannian case. The terms of the answer are computed using ``marching'' operations on permutation diagrams.

math.CO↗

arXiv2003

Gröbner geometry of Schubert polynomials

Our main theorems provide a single geometric setting in which polynomial representatives for Schubert classes in the integral cohomology ring of the flag manifold are determined uniquely, and have positive coefficients for geometric reasons. This results in a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics. Given a permutation w in S_n, we consider a determinantal ideal I_w whose generators are certain minors in the generic n x n matrix (filled with independent variables). Using `multidegrees' as simple algebraic substitutes for torus-equivariant cohomology classes on vector spaces, our main theorems describe, for each ideal I_w: - variously graded multidegrees and Hilbert series in terms of ordinary and double Schubert and Grothendieck polynomials; - a Gröbner basis consisting of minors in the generic n x n matrix; - the Stanley-Reisner complex of the initial ideal in terms of known combinatorial diagrams associated to permutations in S_n; and - a procedure inductive on weak Bruhat order for listing the facets of this complex, thereby generating the coefficients of Schubert polynomials by a positive recursion on combinatorial diagrams. We show that the initial ideal is Cohen-Macaulay, by identifying the Stanley-Reisner complex as a special kind of ``subword complex in S_n'', which we define generally for arbitrary Coxeter groups, and prove to be shellable by giving an explicit vertex decomposition. We also prove geometrically a general positivity statement for multidegrees of subschemes.

math.AG↗

arXiv2003

Subword complexes in Coxeter groups

Let (Π,Σ) be a Coxeter system. An ordered list of elements in Σand an element in Πdetermine a {\em subword complex}, as introduced in our paper on Gröbner geometry of Schubert polynomials (math.AG/0110058). Subword complexes are demonstrated here to be homeomorphic to balls or spheres, and their Hilbert series are shown to reflect combinatorial properties of reduced expressions in Coxeter groups. Two formulae for double Grothendieck polynomials, one of which is due to Fomin and Kirillov, are recovered in the context of simplicial topology for subword complexes. Some open questions related to subword complexes are presented.

math.CO↗

arXiv2003

Four positive formulae for type A quiver polynomials

We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs). Three of our formulae are multiplicity-free and geometric, meaning that their summands have coefficient 1, and correspond bijectively to components of a torus-invariant scheme. The remaining (presently non-geometric) formula was conjectured for by Buch and Fulton in terms of factor sequences of Young tableaux; our proof of it proceeds by way of a new characterization of the tableaux counted by quiver constants. All four formulae come naturally in ``doubled'' versions, two for `double quiver polynomials', and the other two for their stable versions, the `double quiver functions', where setting half the variables equal to the other half specializes to the ordinary case.

math.AG↗

arXiv2003

A Schubert calculus recurrence from the noncomplex W-action on G/B

In this paper, as in our previous "Descent-cycling in Schubert calculus" math.CO/0009112, we study the structure constants in equivariant cohomology of flag manifolds G/B. In this one we give a recurrence (which is frequently, but alas not always, positive) to compute these one by one, using the non-complex action of the Weyl group on G/B. Probably the most noteworthy feature of this recurrence is that to compute a particular structure constant c_{lambda,mu}^nu, one does not have to compute the whole product S_lambda * S_mu.

math.CO↗

arXiv2003

A positive proof of the Littlewood-Richardson rule using the octahedron recurrence

We define the_hive ring_, which has a basis indexed by dominant weights for GL(n), and structure constants given by counting hives [KT1] (or equivalently honeycombs, or Berenstein-Zelevinsky patterns [BZ1]). We use the octahedron rule from [Robbins-Rumsey,Fomin-Zelevinsky,Propp,Speyer] to prove bijectively that this "ring" is indeed associative. This, and the Pieri rule, give a self-contained proof that the hive ring is isomorphic as a ring-with-basis to the representation ring of GL(n). In the honeycomb interpretation, the octahedron rule becomes "scattering" of the honeycombs. This recovers some of the "crosses and wrenches" diagrams from the very recent preprint [S], whose results we use to give a closed form for the associativity bijection.

math.CO↗

arXiv2003

Some schemes related to the commuting variety

The_commuting variety_ is the pairs of NxN matrices (X,Y) such that XY = YX. We introduce the_diagonal commutator scheme_, {(X,Y) : XY-YX is diagonal}, which we prove to be a reduced complete intersection, one component of which is the commuting variety. (We conjecture there to be only one other component.) The diagonal commutator scheme has a flat degeneration to the scheme {(X,Y) : XY lower triangular, YX upper triangular}, which is again a reduced complete intersection, this time with n! components (one for each permutation). The degrees of these components give interesting invariants of permutations.

math.AG↗

arXiv2003

Equivariant K-theory and Equivariant Cohomology

For T an abelian compact Lie group, we give a description of T-equivariant K-theory with complex coefficients in terms of equivariant cohomology. In the appendix we give applications of this by extending results of Chang-Skjelbred and Goresky-Kottwitz-MacPherson from equivariant cohomology to equivariant K-theory.

math.AT↗

arXiv2001

Puzzles and (equivariant) cohomology of Grassmannians

We generalize our puzzle formula for ordinary Schubert calculus on Grassmannians, to a formula for the T-equivariant Schubert calculus. The structure constants to be calculated are polynomials in {y_{i+1} - y_i}; they were shown (abstractly) to have positive coefficients in [Graham] math.AG/9908172. Our formula is the first to be manifestly positive in this sense. In particular this gives a new and self-contained proof of the ordinary puzzle formula, by an induction backwards from the "most equivariant" case. The proof of the formula is mostly combinatorial, but requires no prior combinatorics, and only a modicum of equivariant cohomology (which we include). This formula is closely related to the one in [Molev-Sagan] q-alg/9707028 for multiplying factorial Schur functions in three sets of variables, although their rule does not give a positive formula in the sense of [Graham]. We include a cohomological interpretation of this problem, and a puzzle formulation for it.

math.AT↗

arXiv2001

The honeycomb model of GL(n) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone

The set of possible spectra (λ,μ,ν) of zero-sum triples of Hermitian matrices forms a polyhedral cone. We give a complete determination of its facets, finishing a long story with recent highlights by [Helmke-Rosenthal, Klyachko, Belkale]. We introduce_puzzles_, which are new combinatorial gadgets to compute Grassmannian Schubert calculus, and will probably be the main point of interest for many readers. As the proofs indicate, the Hermitian sum problem is very naturally studied using puzzles directly, and their connection to Schubert calculus is quite incidental to our approach. In particular, we get new, puzzle-theoretic, proofs of the results in [H,Kly,HR,Be]. Along the way we give a characterization of ``rigid'' puzzles, which we use to prove a conjecture of W. Fulton: ``if for a triple of dominant weights λ,μ,νof GL(n,C) the irreducible representation V_νappears exactly once in V_λtensor V_μ, then for all N\in \naturals, V_{Nλ} appears exactly once in V_{Nλ} tensor V_{Nμ}.''

math.CO↗

arXiv2000

Descent-cycling in Schubert calculus

We prove two lemmata about Schubert calculus on generalized flag manifolds G/B, and in the case of the ordinary flag manifold GL_n/B we interpret them combinatorially in terms of descents, and geometrically in terms of missing subspaces. One of them gives a symmetry of Schubert calculus that we christen_descent-cycling_. Computer experiment shows that these lemmata suffice to determine all of GL_n Schubert calculus through n=5, and 99.97%+ at n=6. We use them to give a quick proof of Monk's rule. The lemmata also hold in equivariant (``double'') Schubert calculus for Kac-Moody groups G.

math.CO↗

arXiv2000

Honeycombs and sums of Hermitian matrices

Horn's conjecture, which given the spectra of two Hermitian matrices describes the possible spectra of the sum, was recently settled in the affirmative. In this survey we discuss one of the many steps in this, which required us to introduce a combinatorial gadget called a {\em honeycomb}; the question is then reformulable as about the existence of honeycombs with certain boundary conditions. Another important tool is the connection to the representation theory of the group U(n), by ``classical vs. quantum'' analogies.

math.RT↗

arXiv2000

The symplectic and algebraic geometry of Horn's problem

Horn's problem was the following: given two Hermitian matrices with known spectra, what might be the eigenvalue spectrum of the sum? This linear algebra problem is exactly of the sort to be approached with the methods of modern Hamiltonian geometry (which were unavailable to Horn). The theorem linking symplectic quotients and geometric invariant theory lets one also bring algebraic geometry and representation theory into play. This expository note is intended to elucidate these connections for linear algebraists, in the hope of making it possible to recognize what sort of problems are likely to fall to the same techniques that were used in proving Horn's conjecture.

math.RA↗

arXiv1999

The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture

We introduce the honeycomb model of BZ polytopes, which calculate Littlewood-Richardson coefficients, the tensor product rule for GL(n). Our main result is the existence of a particularly well-behaved honeycomb with given boundary conditions (choice of triple of representations to be tensored together). This honeycomb is necessarily integral, which proves the "saturation conjecture", extending results of Klyachko to give a complete answer to which L-R coefficients are positive. This in turn has as a consequence Horn's conjecture from 1962 characterizing the spectrum of the sum of two Hermitian matrices.

math.RT↗