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Thomas Lam

Publications and source records attributed to Thomas Lam.

At least 73 records · Page 4Linked to original sources
arXiv2014

Grassmannians for scattering amplitudes in 4d $\mathcal{N}=4$ SYM and 3d ABJM

Scattering amplitudes in 4d $\mathcal{N}=4$ super Yang-Mills theory (SYM) can be described by Grassmannian contour integrals whose form depends on whether the external data is encoded in momentum space, twistor space, or momentum twistor space. After a pedagogical review, we present a new, streamlined proof of the equivalence of the three integral formulations. A similar strategy allows us to derive a new Grassmannian integral for 3d $\mathcal{N}=6$ ABJM theory amplitudes in momentum twistor space: it is a contour integral in an orthogonal Grassmannian with the novel property that the internal metric depends on the external data. The result can be viewed as a central step towards developing an amplituhedron formulation for ABJM amplitudes. Various properties of Grassmannian integrals are examined, including boundary properties, pole structure, and a homological interpretation of the global residue theorems for $\mathcal{N}=4$ SYM.

hep-th↗
arXiv2014

Amplituhedron cells and Stanley symmetric functions

The amplituhedron was recently introduced in the study of scattering amplitudes in $N=4$ super Yang-Mills. We compute the cohomology class of a tree amplituhedron subvariety of the Grassmannian to be the truncation of an affine Stanley symmetric function.

math.AG↗
arXiv2014

Infinite reduced words and the Tits boundary of a Coxeter group

Let (W,S) be a finite rank Coxeter system with W infinite. We prove that the limit weak order on the blocks of infinite reduced words of W is encoded by the topology of the Tits boundary of the Davis complex X of W. We consider many special cases, including W word hyperbolic, and X with isolated flats. We establish that when W is word hyperbolic, the limit weak order is the disjoint union of weak orders of finite Coxeter groups. We also establish, for each boundary point ξ, a natural order-preserving correspondence between infinite reduced words which "point towards" ξ, and elements of the reflection subgroup of W which fixes ξ.

math.GR↗
arXiv2014

Intrinsic energy is a loop Schur function

We give an explicit subtraction-free formula for the energy function in tensor products of Kirillov-Reshetikhin crystals for symmetric powers of the standard representation of U_q'(\hat sl_n). The energy function is shown to be the tropicalization of a stretched staircase shape loop Schur function. The latter were introduced by the authors in the study of total positivity in loop groups.

math.QA↗
arXiv2014

Dimers, webs, and positroids

We study the dimer model for a planar bipartite graph N embedded in a disk, with boundary vertices on the boundary of the disk. Counting dimer configurations with specified boundary conditions gives a point in the totally nonnegative Grassmannian. Considering pairing probabilities for the double-dimer model gives rise to Grassmann analogues of Rhoades and Skandera's Temperley-Lieb immanants. The same problem for the (probably novel) triple-dimer model gives rise to the combinatorics of Kuperberg's webs and Grassmann analogues of Pylyavskyy's web immanants. This draws a connection between the square move of plabic graphs (or urban renewal of planar bipartite graphs), and Kuperberg's square reduction of webs. Our results also suggest that canonical-like bases might be applied to the dimer model. We furthermore show that these functions on the Grassmannian are compatible with restriction to positroid varieties. Namely, our construction gives bases for the degree two and degree three components of the homogeneous coordinate ring of a positroid variety that are compatible with the cyclic group action.

math.CO↗
arXiv2014

Whittaker functions, geometric crystals, and quantum Schubert calculus

This mostly expository article explores recent developments in the relations between the three objects in the title from an algebro-combinatorial perspective. We prove a formula for Whittaker functions of a real semisimple group as an integral over a geometric crystal in the sense of Berenstein-Kazhdan. We explain the connections of this formula to the program of mirror symmetry of flag varieties developed by Givental and Rietsch; in particular, the integral formula proves the equivariant version of Rietsch's mirror symmetry conjecture. We also explain the idea that Whittaker functions should be thought of as geometric analogues of irreducible characters of finite-dimensional representations.

math.RT↗
arXiv2013

$k$-Schur functions and affine Schubert calculus

This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculus

math.CO↗
arXiv2012

Linear Laurent phenomenon algebras

In [LP] we introduced Laurent phenomenon algebras, a generalization of cluster algebras. Here we give an explicit description of Laurent phenomenon algebras with a linear initial seed arising from a graph. In particular, any graph associahedron is shown to be the dual cluster complex for some Laurent phenomenon algebra.

math.RT↗
arXiv2012

Loop symmetric functions and factorizing matrix polynomials

These are notes for my talk at ICCM 2010, Beijing. We survey some results, obtained jointly with Pavlo Pylyavskyy, concerning the ring of loop symmetric functions. Motivations from networks on surfaces, total positivity, crystal graphs, and discrete integrable systems are discussed.

math.CO↗
arXiv2012

Box-Basket-Ball Systems

Using the whurl relation of the first two authors, we define a new discrete solitonic system, which we call the box-basket-ball system, generalizing the box-ball system of Takahashi and Satsuma. In box-basket-ball systems balls may be put either into boxes or into baskets. While boxes stay fixed, both balls and baskets get moved during time evolution. Balls and baskets behave as fermionic and bosonic particles respectively. We classify the solitons of this system, and study their scattering.

math.QA↗
arXiv2012

Total positivity, Schubert positivity, and Geometric Satake

Let G be a simple and simply-connected complex algebraic group, and let X \subset G^\vee be the centralizer subgroup of a principal nilpotent element. Ginzburg and Peterson independently related the ring of functions on X with the homology ring of the affine Grassmannian Gr_G. Peterson furthermore connected this ring to the quantum cohomology rings of partial flag varieties G/P. The first aim of this paper is to study three different notions of positivity on X: (1) Schubert positivity arising via Peterson's work, (2) total positivity in the sense of Lusztig, and (3) Mirkovic-Vilonen positivity obtained from the MV-cycles in Gr_G. Our first main theorem establishes that these three notions of positivity coincide. The second aim of this paper is to parametrize the totally nonnegative part of X, confirming a conjecture of the second author. In type A a substantial part of our results were previously established by the second author. The crucial new component of this paper is the connection with the affine Grassmannian and the geometric Satake correspondence.

math.RT↗
arXiv2012

Alcoved Polytopes II

This is the second of two papers where we study polytopes arising from affine Coxeter arrangements. Our results include a formula for their volumes, and also compatible definitions of hypersimplices, descent numbers and major index for all Weyl groups. We give a q-analogue of Weyl's formula for the order of the Weyl group. For A_n, C_n and D_4, we give a Grobner basis which induces the triangulation of alcoved polytopes.

math.CO↗
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

From double quantum Schubert polynomials to k-double Schur functions via the Toda lattice

We show that the k-double Schur functions defined by the authors, and the quantum double Schubert polynomials studied by Kirillov and Maeno and by Ciocan-Fontanine and Fulton, can be obtained from each other by an explicit rational substitution. The main new ingredient is an explicit computation of Kostant's solution to the Toda lattice in terms of equivariant Schubert classes.

math.CO↗
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↗
arXiv2011

Quantum double Schubert polynomials represent Schubert classes

The quantum double Schubert polynomials studied by Kirillov and Maeno, and by Ciocan-Fontanine and Fulton, are shown to represent Schubert classes in Kim's presentation of the equivariant quantum cohomology of the flag variety. We define parabolic analogues of quantum double Schubert polynomials, and show that they represent Schubert classes in the equivariant quantum cohomology of partial flag varieties. For complete flags Anderson and Chen have announced a proof with different methods.

math.CO↗
arXiv2011

Equivariant Pieri Rule for the homology of the affine Grassmannian

An explicit rule is given for the product of the degree two class with an arbitrary Schubert class in the torus-equivariant homology of the affine Grassmannian. In addition a Pieri rule (the Schubert expansion of the product of a special Schubert class with an arbitrary one) is established for the equivariant homology of the affine Grassmannians of SL_n and a similar formula is conjectured for Sp_{2n} and SO_{2n+1}. For SL_n the formula is explicit and positive. By a theorem of Peterson these compute certain products of Schubert classes in the torus-equivariant quantum cohomology of flag varieties. The SL_n Pieri rule is used in our recent definition of k-double Schur functions and affine double Schur functions.

math.CO↗