Allen Knutson

Subword complexes in Coxeter groups

Abstract Let ( Π , Σ ) be a Coxeter system. An ordered list of elements in Σ and an element in Π determine a subword complex , as introduced in Knutson and Miller (Ann. of Math. (2) (2003), to appear). 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 appeared in Fomin and Kirillov (Proceedings of the Sixth Conference in Formal Power Series and Algebraic Combinatorics, DIMACS, 1994, pp. 183–190), are recovered in the context of simplicial topology for subword complexes. Some open questions related to subword complexes are presented.

The symplectic and algebraic geometry of Horn's problem

One version of Horns problem asks for which λ,μ,ν does Hλ+Hμ+Hν=0 have solutions, where Hλ,μ,ν are Hermitian matrices with spectra λ,μ,ν. This turns out to be a moment map condition in Hamiltonian geometry. Many of the results around Horns problem proven with great effort “by hand” are in fact simple consequences of the modern machinery of symplectic geometry, and the subtler ones provable via the connection to geometric invariant theory. We give an overview of this theory (which was not available to Horn), including all definitions, and how it can be used in linear algebra.

Gröbner geometry of vertex decompositions and of flagged tableaux

Abstract We relate a classic algebro-geometric degeneration technique, dating at least to Hodge 1941 (J. London Math. Soc. 16: 245–255), 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 (Discr. Math. 153: 123–143), 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 (J. Combin. Th. (A) 40: 276–289) on flagged tableaux, and Buch 2002 (Acta. Math. 189: 37–78) on set-valued tableaux, giving geometric meaning to both. This work focuses on diagonal term orders, giving results complementary to those of Knutson-Miller 2005 (Ann. Math. 161: 1245–1318), 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.

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.

Complete moduli spaces of branchvarieties

Abstract 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 diagonal, 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.

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 GLn/ 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 these two lemmata are surprisingly powerful: they already suffice to determine all of GLn Schubert calculus through n = 5, and 99.97%+ at n = 6. We use them to give a quick proof of Monks rule. The lemmata also hold in equivariant (“double”) Schubert calculus for Kac– Moody groups G.

A formula for K-theory truncation schubert calculus

Author(s): Knutson, A; Yong, Alexander T | Abstract: Define a truncation rt (p) of a polynomial p in { x1,x2,x3,…} 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 formulas for certain Schubert structure constants in K ( Flags ( C^n)), in particular generalizing those studied by Kogan (2001) in which only cohomology was treated, and those studied by Buch (2002) on the Grassmannian case. The terms in the answer are computed using “marching” operations on permutation diagrams.

A limit of toric symplectic forms that has no periodic Hamiltonians

Abstract. We calculate the Riemann-Roch number of some of the pentagon spaces defined in [Kl], [KM], [HK1]. Using this, we show that while the regular pentagon space is diffeomorphic to a toric variety, even symplectomorphic to one under arbitrarily small perturbations of its symplectic structure, it does not admit a symplectic circle action. In particular, within the cohomology classes of symplectic structures, the subset admitting a circle action is not closed.

Cohomology rings of symplectic cuts

Abstract The “symplectic cut” construction of Lerman produces two symplectic orbifolds C − and C + from a symplectic manifold M with a Hamiltonian circle action. We compute the rational cohomology ring of C + in terms of those of M and C − .

The Multidegree of the Multi-Image Variety

The multi-image variety is a subvariety of \(\mathop{\mathrm{Gr}}\nolimits (1, \mathbb{P}^{3})^{n}\) that parametrizes all of the possible images that can be taken by n fixed cameras. We compute its cohomology class in the cohomology ring of \(\mathop{\mathrm{Gr}}\nolimits (1, \mathbb{P}^{3})^{n}\) and its multidegree as a subvariety of \((\mathbb{P}^{5})^{n}\) under the Plucker embedding.