DOCSLIB.ORG

Constructing Packings in Grassmannian Manifolds Via Alternating Projection

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Constructing Packings in Grassmannian Manifolds Via Alternating Projection Constructing Packings in Grassmannian Manifolds via Alternating Projection I. S. Dhillon, R. W. Heath Jr., T. Strohmer, and J. A. Tropp CONTENTS This paper describes a numerical method for finding good pack- 1. Introduction ings in Grassmannian manifolds equipped with various met- 2. Packing in Grassmannian Manifolds rics. This investigation also encompasses packing in projective 3. Alternating Projection for Chordal Distance spaces. In each case, producing a good packing is equivalent 4. Bounds on the Packing Diameter to constructing a matrix that has certain structural and spectral 5. Experiments properties. By alternately enforcing the structural condition and 6. Discussion then the spectral condition, it is often possible to reach a matrix 7. Tammes’ Problem that satisfies both. One may then extract a packing from this Acknowledgments matrix. References This approach is both powerful and versatile. In cases in which experiments have been performed, the alternating projection method yields packings that compete with the best packings recorded. It also extends to problems that have not been studied numerically. For example, it can be used to produce packings of subspaces in real and complex Grassmannian spaces equipped with the Fubini–Study distance; these packings are valuable in wireless communications. One can prove that some of the novel configurations constructed by the algorithm have packing diam- eters that are nearly optimal. 1. INTRODUCTION Let us begin with the standard facetious example. Imag- ine that several mutually inimical nations build their cap- ital cities on the surface of a featureless globe. Being concerned about missile strikes, they wish to locate the closest pair of cities as far apart as possible. In other words, what is the best way to pack points on the sur- face of a two-dimensional sphere? This question, first discussed by the Dutch biologist Tammes [Tammes 30], is the prototypical example of packing in a compact metric space. It has been stud- ied in detail for the last 75 years. More recently, re- 2000 AMS Subject Classification: Primary 51N15, 52C17 searchers have started to ask about packings in other compact spaces. In particular, several communities have Keywords: Combinatorial optimization, packing, projective spaces, Grassmannian spaces, Tammes’ Problem investigated how to arrange subspaces in a Euclidean c A K Peters, Ltd. 1058-6458/2008 $ 0.50 per page Experimental Mathematics 17:1, page 9 10 Experimental Mathematics, Vol. 17 (2008), No. 1 space so that they are as distinct as possible. An equiv- 1.2 Alternating Projection alent formulation is to find the best packings of points We will attempt to solve the feasibility problem (1–1) in a Grassmannian manifold. This problem has appli- in Grassmannian manifolds equipped with a number of cations in quantum computing and wireless communica- different metrics, but the same basic algorithm applies tions. There has been theoretical interest in subspace in each case. Here is a high-level description of our ap- packing since the 1960s [T´oth 65], but the first detailed proach. numerical study appears in a 1996 paper of Conway, First, we show that each configuration of subspaces is Hardin, and Sloane [Conway et al. 96]. associated with a block Gram matrix whose blocks con- The aim of this paper is to describe a flexible numer- trol the distances between pairs of subspaces. Then we ical method that can be used to construct packings in prove that a configuration solves the feasibility problem Grassmannian manifolds equipped with several different (1–1) if and only if its Gram matrix possesses both a metrics. The rest of this introduction provides a formal structural property and a spectral property. The overall statement of abstract packing problems, and it offers an algorithm consists of the following steps. overview of our approach to solving them. 1. Choose an initial configuration and construct its ma- 1.1 Abstract Packing Problems trix. Although we will be working with Grassmannian mani- 2. Alternately enforce the structural condition and the folds, it is more instructive to introduce packing problems spectral condition in hope of reaching a matrix that M in an abstract setting. Let be a compact metric space satisfies both. endowed with the distance function distM.Thepacking diameter of a finite subset X is the minimum distance be- 3. Extract a configuration of subspaces from the output tween some pair of distinct points drawn from X . That matrix. is, def In our work, we choose the initial configuration ran- packM(X ) = min distM(xm,xn). m=n domly and then remove similar subspaces from it with a simple algorithm. One can imagine more sophisticated In other words, the packing diameter of a set is the diam- approaches to constructing the initial configuration. eter of the largest open ball that can be centered at each Flexibility and ease of implementation are the major point of the set without encompassing any other point. advantages of alternating projection. This article demon- (It is also common to study the packing radius,whichis strates that appropriate modifications of this basic tech- half the diameter of this ball.) An optimal packing of nique allow us to construct solutions to the feasibility N X points is an ensemble that solves the mathematical problem in Grassmannian manifolds equipped with var- program ious metrics. Some of these problems have never been max packM(X ), studied numerically, and the experiments point toward |X |=N intriguing phenomena that deserve theoretical attention. where |·| returns the cardinality of a finite set. The op- Moreover, we believe that the possibilities of this method timal packing problem is guaranteed to have a solution have not been exhausted and that it will see other appli- because the metric space is compact and the objective is cations in the future. a continuous function of the ensemble X . Alternating projection does have several drawbacks. This article focuses on a feasibility problem closely con- It may converge very slowly, and it does not always yield nected with optimal packing. Given a number ρ, the goal a high level of numerical precision. In addition, it may is to produce a set of N points for which not deliver good packings when the ambient dimension or the number of subspaces in the configuration is large. packM(X ) ≥ ρ. (1–1) 1.3 Motivation and Related Work This problem is notoriously difficult to solve because it This work was motivated by applications in electrical en- is highly nonconvex, and it is even more difficult to de- gineering. In particular, subspace packings solve certain termine the maximum value of ρ for which the feasibility extremal problems that arise in multiple-antenna com- problem is soluble. This maximum value of ρ corresponds munication systems [Zheng and Tse 02, Hochwald et al. to the diameter of an optimal packing. 00, Love et al. 04]. This application requires complex Dhillon et al.: Constructing Packings in Grassmannian Manifolds via Alternating Projection 11 Grassmannian packings that consist of a small number 1.5 Outline of Article of subspaces in an ambient space of low dimension. Our Here is a brief overview of this article. In Section 2, we algorithm is quite effective in this parameter regime. The develop a basic description of Grassmannian manifolds resulting packings fill a significant gap in the literature, and present some natural metrics. Section 3 explains since existing tables consider only the real case [Sloane why alternating projection is a natural algorithm for pro- 04a]. See Section 6.1 for additional discussion of the wire- ducing Grassmannian packings, and it outlines how to less application. apply this algorithm for one specific metric. Section 4 The approach to packing via alternating projection gives some theoretical upper bounds on the optimal di- was discussed in a previous publication [Tropp et al. ameter of packings in Grassmannian manifolds. Section 05], but the experiments were limited to a single case. 5 describes the outcomes of an extensive set of numeri- We are aware of several other numerical methods that cal experiments and explains how to apply the algorithm can be used to construct packings in Grassmannian to other metrics. Section 6 offers some discussion and manifolds [Conway et al. 96, Trosset 01, Agrawal et conclusions. Appendix 7 explores how our methodology al. 01]. These techniques rely on ideas from nonlinear applies to Tammes’ problem of packing on the surface of programming. a sphere. Our experiments resulted in tables of packing diame- ters. We did not store the configurations produced by the 1.4 Historical Interlude algorithm. The Matlab code that produced these data is The problem of constructing optimal packings in various available on request from [email protected]. metric spaces has a long and lovely history. The most These tables and figures are intended only to describe famous example may be Kepler’s conjecture that an op- the results of our experiments; it is likely that many of the timal packing of spheres in three-dimensional Euclidean packing diameters could be improved with additional ef- space1 locates them at the points of a face-centered cu- fort. In all cases, we present the results of calculations for bic lattice. For millennia, greengrocers have applied this the stated problem, even if we obtained a better packing theorem when stacking oranges, but it has only been by solving a different problem. For example, a complex established rigorously within the last few years [Hales packing should always improve on the corresponding real 04]. Packing problems play a major role in modern com- packing. If the numbers indicate otherwise, it just means munications because error-correcting codes may be in- that the complex experiment yielded an inferior result. terpreted as packings in the Hamming space of binary As a second example, the optimal packing diameter must strings [Cover and Thomas 91].
Load more

Recommended publications
  • Classification on the Grassmannians
    GEOMETRIC FRAMEWORK SOME EMPIRICAL RESULTS COMPRESSION ON G(k, n) CONCLUSIONS Classification on the Grassmannians: Theory and Applications Jen-Mei Chang Department of Mathematics and Statistics California State University, Long Beach [email protected] AMS Fall Western Section Meeting Special Session on Metric and Riemannian Methods in Shape Analysis October 9, 2010 at UCLA GEOMETRIC FRAMEWORK SOME EMPIRICAL RESULTS COMPRESSION ON G(k, n) CONCLUSIONS Outline 1 Geometric Framework Evolution of Classification Paradigms Grassmann Framework Grassmann Separability 2 Some Empirical Results Illumination Illumination + Low Resolutions 3 Compression on G(k, n) Motivations, Definitions, and Algorithms Karcher Compression for Face Recognition GEOMETRIC FRAMEWORK SOME EMPIRICAL RESULTS COMPRESSION ON G(k, n) CONCLUSIONS Architectures Historically Currently single-to-single subspace-to-subspace )2( )3( )1( d ( P,x ) d ( P,x ) d(P,x ) ... P G , , … , x )1( x )2( x( N) single-to-many many-to-many Image Set 1 … … G … … p * P , Grassmann Manifold Image Set 2 * X )1( X )2( … X ( N) q Project Eigenfaces GEOMETRIC FRAMEWORK SOME EMPIRICAL RESULTS COMPRESSION ON G(k, n) CONCLUSIONS Some Approaches • Single-to-Single 1 Euclidean distance of feature points. 2 Correlation. • Single-to-Many 1 Subspace method [Oja, 1983]. 2 Eigenfaces (a.k.a. Principal Component Analysis, KL-transform) [Sirovich & Kirby, 1987], [Turk & Pentland, 1991]. 3 Linear/Fisher Discriminate Analysis, Fisherfaces [Belhumeur et al., 1997]. 4 Kernel PCA [Yang et al., 2000]. GEOMETRIC FRAMEWORK SOME EMPIRICAL RESULTS COMPRESSION ON G(k, n) CONCLUSIONS Some Approaches • Many-to-Many 1 Tangent Space and Tangent Distance - Tangent Distance [Simard et al., 2001], Joint Manifold Distance [Fitzgibbon & Zisserman, 2003], Subspace Distance [Chang, 2004].
    [Show full text]
  • Arxiv:1711.05949V2 [Math.RT] 27 May 2018 Mials, One Obtains a Sum Whose Summands Are Rational Functions in Ti
    RESIDUES FORMULAS FOR THE PUSH-FORWARD IN K-THEORY, THE CASE OF G2=P . ANDRZEJ WEBER AND MAGDALENA ZIELENKIEWICZ Abstract. We study residue formulas for push-forward in the equivariant K-theory of homogeneous spaces. For the classical Grassmannian the residue formula can be obtained from the cohomological formula by a substitution. We also give another proof using sym- plectic reduction and the method involving the localization theorem of Jeffrey–Kirwan. We review formulas for other classical groups, and we derive them from the formulas for the classical Grassmannian. Next we consider the homogeneous spaces for the exceptional group G2. One of them, G2=P2 corresponding to the shorter root, embeds in the Grass- mannian Gr(2; 7). We find its fundamental class in the equivariant K-theory KT(Gr(2; 7)). This allows to derive a residue formula for the push-forward. It has significantly different character comparing to the classical groups case. The factor involving the fundamen- tal class of G2=P2 depends on the equivariant variables. To perform computations more efficiently we apply the basis of K-theory consisting of Grothendieck polynomials. The residue formula for push-forward remains valid for the homogeneous space G2=B as well. 1. Introduction ∗ r Suppose an algebraic torus T = (C ) acts on a complex variety X which is smooth and complete. Let E be an equivariant complex vector bundle over X. Then E defines an element of the equivariant K-theory of X. In our setting there is no difference whether one considers the algebraic K-theory or the topological one.
    [Show full text]
  • Filtering Grassmannian Cohomology Via K-Schur Functions
    FILTERING GRASSMANNIAN COHOMOLOGY VIA K-SCHUR FUNCTIONS THE 2020 POLYMATH JR. \Q-B-AND-G" GROUPy Abstract. This paper is concerned with studying the Hilbert Series of the subalgebras of the cohomology rings of the complex Grassmannians and La- grangian Grassmannians. We build upon a previous conjecture by Reiner and Tudose for the complex Grassmannian and present an analogous conjecture for the Lagrangian Grassmannian. Additionally, we summarize several poten- tial approaches involving Gr¨obnerbases, homological algebra, and algebraic topology. Then we give a new interpretation to the conjectural expression for the complex Grassmannian by utilizing k-conjugation. This leads to two conjectural bases that would imply the original conjecture for the Grassman- nian. Finally, we comment on how this new approach foreshadows the possible existence of analogous k-Schur functions in other Lie types. Contents 1. Introduction2 2. Conjectures3 2.1. The Grassmannian3 2.2. The Lagrangian Grassmannian4 3. A Gr¨obnerBasis Approach 12 3.1. Patterns, Patterns and more Patterns 12 3.2. A Lex-greedy Approach 20 4. An Approach Using Natural Maps Between Rings 23 4.1. Maps Between Grassmannians 23 4.2. Maps Between Lagrangian Grassmannians 24 4.3. Diagrammatic Reformulation for the Two Main Conjectures 24 4.4. Approach via Coefficient-Wise Inequalities 27 4.5. Recursive Formula for Rk;` 27 4.6. A \q-Pascal Short Exact Sequence" 28 n;m 4.7. Recursive Formula for RLG 30 4.8. Doubly Filtered Basis 31 5. Schur and k-Schur Functions 35 5.1. Schur and Pieri for the Grassmannian 36 5.2. Schur and Pieri for the Lagrangian Grassmannian 37 Date: August 2020.
    [Show full text]
  • §4 Grassmannians 73
    §4 Grassmannians 4.1 Plücker quadric and Gr(2,4). Let 푉 be 4-dimensional vector space. e set of all 2-di- mensional vector subspaces 푈 ⊂ 푉 is called the grassmannian Gr(2, 4) = Gr(2, 푉). More geomet- rically, the grassmannian Gr(2, 푉) is the set of all lines ℓ ⊂ ℙ = ℙ(푉). Sending 2-dimensional vector subspace 푈 ⊂ 푉 to 1-dimensional subspace 훬푈 ⊂ 훬푉 or, equivalently, sending a line (푎푏) ⊂ ℙ(푉) to 한 ⋅ 푎 ∧ 푏 ⊂ 훬푉 , we get the Plücker embedding 픲 ∶ Gr(2, 4) ↪ ℙ = ℙ(훬 푉). (4-1) Its image consists of all decomposable¹ grassmannian quadratic forms 휔 = 푎∧푏 , 푎, 푏 ∈ 푉. Clearly, any such a form has zero square: 휔 ∧ 휔 = 푎 ∧ 푏 ∧ 푎 ∧ 푏 = 0. Since an arbitrary form 휉 ∈ 훬푉 can be wrien¹ in appropriate basis of 푉 either as 푒 ∧ 푒 or as 푒 ∧ 푒 + 푒 ∧ 푒 and in the laer case 휉 is not decomposable, because of 휉 ∧ 휉 = 2 푒 ∧ 푒 ∧ 푒 ∧ 푒 ≠ 0, we conclude that 휔 ∈ 훬 푉 is decomposable if an only if 휔 ∧ 휔 = 0. us, the image of (4-1) is the Plücker quadric 푃 ≝ { 휔 ∈ 훬푉 | 휔 ∧ 휔 = 0 } (4-2) If we choose a basis 푒, 푒, 푒, 푒 ∈ 푉, the monomial basis 푒 = 푒 ∧ 푒 in 훬 푉, and write 푥 for the homogeneous coordinates along 푒, then straightforward computation 푥 ⋅ 푒 ∧ 푒 ∧ 푥 ⋅ 푒 ∧ 푒 = 2 푥 푥 − 푥 푥 + 푥 푥 ⋅ 푒 ∧ 푒 ∧ 푒 ∧ 푒 < < implies that 푃 is given by the non-degenerated quadratic equation 푥푥 = 푥푥 + 푥푥 .
    [Show full text]
  • 8. Grassmannians
    66 Andreas Gathmann 8. Grassmannians After having introduced (projective) varieties — the main objects of study in algebraic geometry — let us now take a break in our discussion of the general theory to construct an interesting and useful class of examples of projective varieties. The idea behind this construction is simple: since the definition of projective spaces as the sets of 1-dimensional linear subspaces of Kn turned out to be a very useful concept, let us now generalize this and consider instead the sets of k-dimensional linear subspaces of Kn for an arbitrary k = 0;:::;n. Definition 8.1 (Grassmannians). Let n 2 N>0, and let k 2 N with 0 ≤ k ≤ n. We denote by G(k;n) the set of all k-dimensional linear subspaces of Kn. It is called the Grassmannian of k-planes in Kn. Remark 8.2. By Example 6.12 (b) and Exercise 6.32 (a), the correspondence of Remark 6.17 shows that k-dimensional linear subspaces of Kn are in natural one-to-one correspondence with (k − 1)- n− dimensional linear subspaces of P 1. We can therefore consider G(k;n) alternatively as the set of such projective linear subspaces. As the dimensions k and n are reduced by 1 in this way, our Grassmannian G(k;n) of Definition 8.1 is sometimes written in the literature as G(k − 1;n − 1) instead. Of course, as in the case of projective spaces our goal must again be to make the Grassmannian G(k;n) into a variety — in fact, we will see that it is even a projective variety in a natural way.
    [Show full text]
  • Grassmannians Via Projection Operators and Some of Their Special Submanifolds ∗
    GRASSMANNIANS VIA PROJECTION OPERATORS AND SOME OF THEIR SPECIAL SUBMANIFOLDS ∗ IVKO DIMITRIC´ Department of Mathematics, Penn State University Fayette, Uniontown, PA 15401-0519, USA E-mail: [email protected] We study submanifolds of a general Grassmann manifold which are of 1-type in a suitably defined Euclidean space of F−Hermitian matrices (such a submanifold is minimal in some hypersphere of that space and, apart from a translation, the im- mersion is built using eigenfunctions from a single eigenspace of the Laplacian). We show that a minimal 1-type hypersurface of a Grassmannian is mass-symmetric and has the type-eigenvalue which is a specific number in each dimension. We derive some conditions on the mean curvature of a 1-type hypersurface of a Grassmannian and show that such a hypersurface with a nonzero constant mean curvature has at most three constant principal curvatures. At the end, we give some estimates of the first nonzero eigenvalue of the Laplacian on a compact submanifold of a Grassmannian. 1. Introduction The Grassmann manifold of k dimensional subspaces of the vector space − Fm over a field F R, C, H can be conveniently defined as the base man- ∈{ } ifold of the principal fibre bundle of the corresponding Stiefel manifold of F unitary frames, where a frame is projected onto the F plane spanned − − by it. As observed already in 1927 by Cartan, all Grassmannians are sym- metric spaces and they have been studied in such context ever since, using the root systems and the Lie algebra techniques. However, the submani- fold geometry in Grassmannians of higher rank is much less developed than the corresponding geometry in rank-1 symmetric spaces, the notable excep- tions being the classification of the maximal totally geodesic submanifolds (the well known results of J.A.
    [Show full text]
  • WEIGHT FILTRATIONS in ALGEBRAIC K-THEORY Daniel R
    November 13, 1992 WEIGHT FILTRATIONS IN ALGEBRAIC K-THEORY Daniel R. Grayson University of Illinois at Urbana-Champaign Abstract. We survey briefly some of the K-theoretic background related to the theory of mixed motives and motivic cohomology. 1. Introduction. The recent search for a motivic cohomology theory for varieties, described elsewhere in this volume, has been largely guided by certain aspects of the higher algebraic K-theory developed by Quillen in 1972. It is the purpose of this article to explain the sense in which the previous statement is true, and to explain how it is thought that the motivic cohomology groups with rational coefficients arise from K-theory through the intervention of the Adams operations. We give a basic description of algebraic K-theory and explain how Quillen’s idea [42] that the Atiyah-Hirzebruch spectral sequence of topology may have an algebraic analogue guides the search for motivic cohomology. There are other useful survey articles about algebraic K-theory: [50], [46], [23], [56], and [40]. I thank W. Dwyer, E. Friedlander, S. Lichtenbaum, R. McCarthy, S. Mitchell, C. Soul´e and R. Thomason for frequent and useful consultations. 2. Constructing topological spaces. In this section we explain the considerations from combinatorial topology which give rise to the higher algebraic K-groups. The first principle is simple enough to state, but hard to implement: when given an interesting group (such as the Grothendieck group of a ring) arising from some algebraic situation, try to realize it as a low-dimensional homotopy group (especially π0 or π1) of a space X constructed in some combinatorial way from the same algebraic inputs, and study the homotopy type of the resulting space.
    [Show full text]
  • Equivariant Homology and K-Theory of Affine Grassmannians and Toda Lattices R Bezrukavnikov
    University of Massachusetts Amherst ScholarWorks@UMass Amherst Mathematics and Statistics Department Faculty Mathematics and Statistics Publication Series 2005 Equivariant homology and K-theory of affine Grassmannians and Toda lattices R Bezrukavnikov M Finkelberg I Mirkovic University of Massachusetts - Amherst, [email protected] Follow this and additional works at: https://scholarworks.umass.edu/math_faculty_pubs Part of the Physical Sciences and Mathematics Commons Recommended Citation Bezrukavnikov, R; Finkelberg, M; and Mirkovic, I, "Equivariant homology and K-theory of affine asGr smannians and Toda lattices" (2005). COMPOSITIO MATHEMATICA. 727. Retrieved from https://scholarworks.umass.edu/math_faculty_pubs/727 This Article is brought to you for free and open access by the Mathematics and Statistics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Mathematics and Statistics Department Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. EQUIVARIANT (K-)HOMOLOGY OF AFFINE GRASSMANNIAN AND TODA LATTICE ROMAN BEZRUKAVNIKOV, MICHAEL FINKELBERG, AND IVAN MIRKOVIC´ 1. Introduction 1.1. Let G be an almost simple complex algebraic group, and let GrG be its affine Grassmannian. Recall that if we set O = C[[t]], F = C((t)), then GrG = G(F)/G(O). It is well-known that the subgroup ΩK of polynomial loops into a maximal compact subgroup K G projects isomorphically to GrG; thus GrG acquires the structure of a topological⊂ group. An algebro-geometric counterpart of this structure is provided by the convolution diagram G(F) O Gr Gr . ×G( ) G → G It allows one to define the convolution of two G(O) equivariant geometric objects (such as sheaves, or constrictible functions) on GrG.
    [Show full text]
  • Cohomology of the Complex Grassmannian
    COHOMOLOGY OF THE COMPLEX GRASSMANNIAN JONAH BLASIAK Abstract. The Grassmannian is a generalization of projective spaces–instead of looking at the set of lines of some vector space, we look at the set of all n-planes. It can be given a manifold structure, and we study the cohomology ring of the Grassmannian manifold in the case that the vector space is complex. The mul- tiplicative structure of the ring is rather complicated and can be computed using the fact that for smooth oriented manifolds, cup product is Poincar´edual to intersection. There is some nice com- binatorial machinery for describing the intersection numbers. This includes the symmetric Schur polynomials, Young tableaux, and the Littlewood-Richardson rule. Sections 1, 2, and 3 introduce no- tation and the necessary topological tools. Section 4 uses linear algebra to prove Pieri’s formula, which describes the cup product of the cohomology ring in a special case. Section 5 describes the combinatorics and algebra that allow us to deduce all the multi- plicative structure of the cohomology ring from Pieri’s formula. 1. Basic properties of the Grassmannian The Grassmannian can be defined for a vector space over any field; the cohomology of the Grassmannian is the best understood for the complex case, and this is our focus. Following [MS], the complex Grass- m+n mannian Gn(C ) is the set of n-dimensional complex linear spaces, or n-planes for brevity, in the complex vector space Cm+n topologized n+m as follows: Let Vn(C ) denote the subspace of the n-fold direct sum Cm+n ⊕ ..
    [Show full text]
  • Equivariant Homology and K -Theory of Affine Grassmannians and Toda
    Compositio Math. 141 (2005) 746–768 DOI: 10.1112/S0010437X04001228 Equivariant homology and K-theory of affine Grassmannians and Toda lattices Roman Bezrukavnikov, Michael Finkelberg and Ivan Mirkovi´c Dedicated to Vladimir Drinfeld on the occasion of his 50th birthday Abstract For an almost simple complex algebraic group G with affine Grassmannian GrG = G(C[[t]]) G(C((t)))/G(C[[t]]), we consider the equivariant homology H (GrG)andK-theory G(C[[t]]) K (GrG). They both have a commutative ring structure with respect to convolution. We identify the spectrum of homology ring with the universal group-algebra centralizer of the Langlands dual group Gˇ, and we relate the spectrum of K-homology ring to the universal group-group centralizer of G and of Gˇ. If we add the loop-rotation equivariance, we obtain a noncommutative deformation of the (K-)homology ring, and thus a Poisson structure on its spectrum. We relate this structure to the standard one on the universal centralizer. The commutative subring of G(C[[t]])-equivariant homology of the point gives rise to a polarization which is related to Kostant’s Toda lattice integrable system. We also compute the equivariant K-ring of the affine Grassmannian Steinberg variety. The equivariant K-homology of GrG is equipped with a canonical basis formed by the classes of simple equivariant perverse coherent sheaves. Their convolution is again perverse and is related to the Feigin–Loktev fusion product of G(C[[t]])-modules. 1. Introduction Let G be an almost simple complex algebraic group, and let GrG be its affine Grassmannian.
    [Show full text]
  • MATH 465/565: Grassmannian Notes
    MATH 465/565: Grassmannian Notes The Grassmannian G(r; n) is the set of r-dimensional subspaces of the k-vector space kn; it has a natural bijection with the set G(r − 1; n − 1) of (r − 1)-dimensional linear subspaces Pr−1 ⊆ Pn. We write G(k; V ) for the set of k-dimensional subspaces of an n-dimensional k-vector space V . We'd like to be able to think of G(r; V ) as a quasiprojective variety; to do so, we consider the Pl¨uckerembedding: r γ ∶ G(r; V ) → P V Span(v1; : : : ; vr) ↦ [v1 ∧ ⋯ ∧ vr] If w a v is another ordered basis for Λ Span v ; : : : ; v , where A a is an invertible i = ∑j ij j1≤i≤r = ( 1 r) = ( ij) matrix, then w1 ∧⋯∧wr = (det A)(v1 ∧⋯∧vr). Thus the Pl¨ucker embedding is a well-defined function r from G(k; V ) to P ⋀ V . We would like to show, in analogy with what we were able to show for the Segre embedding σ∶ P(V ) × P(W ) → P(V ⊗ W ), that • the Pl¨ucker embedding γ is injective, • the image γ(G(r; V )) is closed, and • the Grassmannian G(r; V ) \locally" can be given a structure as an affine variety, and γ restricts to an isomorphism between these \local" pieces of G(r; V ) and Zariski open subsets of the image. r Given x ∈ ⋀ V , we say that x is totally decomposable if x = v1 ∧ ⋯ ∧ vr for some v1; : : : ; vr ∈ V , or equivalently, if [x] is in the image of the Pl¨ucker embedding.
    [Show full text]
  • The Grassmannian
    The Grassmannian Sam Panitch under the mentorship of Professor Chris Woodward and Marco Castronovo supported by the Rutgers Math Department June 11, 2020 DIMACS REU at Rutgers University 1 Definition Let V be an n-dimensional vector space, and fix an integer d < n • The Grassmannian, denoted Grd;V is the set of all d-dimensional vector subspaces of V • This is a manifold and a variety 2 Some Simple Examples For very small d and n, the Grassmannian is not very interesting, but it may still be enlightening to explore these examples in Rn 1. Gr1;2 - All lines in a 2D space ! P 2 2. Gr1;3 - P 3. Gr2;3 - we can identify each plane through the origin with a 2 unique perpendicular line that goes through the origin ! P 3 The First Interesting Grassmannian Let's spend some time exploring Gr2;4, as it turns out this the first Grassmannian over Euclidean space that is not just a projective space. • Consider the space of rank 2 (2 × 4) matrices with A ∼ B if A = CB where det(C) > 0 • Let B be a (2 × 4) matrix. Let Bij denote the minor from the ith and jth column. A simple computation shows B12B34 − B13B24 + B14B23 = 0 5 • Let Ω ⊂ P be such that (x; y; z; t; u; v) 2 Ω () xy − zt + uv = 0: Define a map 5 f : M(2 × 4) ! Ω ⊂ P , f (B) = (B12; B34; B13; B24; B14B23). It can be shown this is a bijection 4 The First Interesting Grassmannian Continued... Note for any element in Ω, and we can consider (1; y; z; t; u; v) and note that the matrix " # 1 0 −v −t 0 1 z u will map to this under f , showing we really only need 4 parameters, i.e, the dimension is k(n − k) = 2(4 − 2) = 4 We hope to observe similar trends in the more general case 5 An Atlas for the Grassmannian We will now show that Grk;V is a smooth manifold of dimension k(n − k).
    [Show full text]