Research Statement. Edward Richmond. October 13, 2012

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1 Research Statement Edward Richmond October 13, 2012 Introduction My mathematical interests include algebraic combinatorics, algebraic geometry and Lie theory. In particular, I study Schubert calculus, flag varieties, Coxeter groups and their related applications. In this statement, I discuss four topics on different aspects of my research. The first topic involves two applications of Schubert calculus of the Grassmannian to problems on eigenvalues of hermitian matrices. The second topic is on the combinatorics of Coxeter groups and the detection of palindromic Poincaré polynomials. The third and fourth topics deal with the development of computational tools for computing Schubert structure constants using techniques in algebra and geometry respectively. In the last section, I discuss some research projects currently in progress. 1 Littlewood-Richardson coefficients This section is about two projects involving Schubert calculus of the Grassmannian Gr(r, n) of r-dimensional subspaces in C n. The cohomology ring H (Gr(r, n)) has an additive basis of Schubert classes {σ λ } λ Λ, where Λ is the set of partitions whose Young diagrams are contained in an r (n r) rectangle. For any three partitions λ, µ, ν Λ we can define the Littlewood- Richardson coefficients c ν λ,µ by the product structure constants σ λ σ µ = ν Λ c ν λ,µ σ ν. The Littlewood-Richardson coefficients arise in several fields of mathematics including the representation theory of the general linear group, the combinatorics of symmetric functions, and quiver representations. One remarkable application of Littlewood-Richardson coefficients is to the eigenvalue problem on sums of hermitian matrices. The following theorem is proved by the combined works of Klyachko [12] and Knutson and Tao[13]. Theorem 1. ([12, 13]) The coefficient c ν λ,µ > 0 if and only if there exist r r hermitian matrices A, B, C with eigenvalues given by the partitions λ, µ, ν and A + B = C. In joint work with D. Anderson and A. Yong [1], we are able to extend this result to the setting of torus-equivariant cohomology of the Grassmannian HT (Gr(r, n)). Define the structure constants 1

2 Cλ,µ ν by the product of equivariant Schubert classes Σ λ Σ µ = ν Λ C ν λ,µ Σ ν. We have the following theorem (omitting some technical constraints). Theorem 2. ([1]) The coefficient Cλ,µ ν > 0 if and only if there exist r r hermitian matrices A, B, C with eigenvalues given by the partitions λ, µ, ν and A + B C. Here a matrix A B if A B is positive semi-definite. Theorem 2 is proved by showing that Horn s inequalities, which determine when c ν λ,µ > 0, also determine when Cν λ,µ > 0 in the equivariant setting. As a corollary, we get an equivariant generalization of the celebrated saturation theorem. Theorem 3. ([1]) C ν λ,µ > 0 if and only if CNν Nλ,Nµ > 0 for any N > 0. Another application of Theorem 1 is to frame theory, an important topic in functional analysis. Let P 1,..., P k be a sequence of N N orthogonal projection matrices and let L := (L 1,..., L k ) denote the corresponding rank sequence (i.e. rk(p i ) = L i ). We say that P 1,..., P k is a tight fusion frame if there exists a real number α such that k P i = αi i=1 where I denotes the identity matrix. Applications of fusion frames include sensor networks [9], coding theory [5, 15], compressed sensing [6] and filter banks [10]. In [7], together with M. Bownik and K. Luoto, we address the problem of classifying all L that are rank sequences of some tight fusion frame. Since orthogonal projection matrices are hermitian, we use Theorem 1 to prove the following classification. Theorem 4. ([7]) L = (L 1,..., L k ) is a tight fusion frame sequence if and only if k σ (N L i) 0 i=1 in H (Gr(N, M + N)) where M := k i=1 L i and the partition (N L i ) := (N,..., N) }{{} L i. This connection between frame theory and Schubert calculus yields many interesting results in both fields of mathematics. For example, using Schubert combinatorics, we produce new bounding estimates on tight fusion frames previously unknown in frame theory. Conversely, inspired by dualities found in frame theory, we construct new combinatorial identities for Littlewood- Richardson coefficients. 2

3 2 Coxeter groups and Poincaré polynomials A Coxeter group W is a group generated by a finite set S subject to the relations s 2 = 1 and (st) mst = 1 for s, t S and m st {2, 3,..., }. Important examples of Coxeter groups include permutation groups, dihedral groups, triangle groups and Weyl groups. Coxeter groups come equipped with a length function l and Bruhat partial order. For any w W we can define the Poincaré polynomial P w (q) := u w q l(u). In [21], W. Slofstra and I study the problem of determining when P w (q) is a palindromic polynomial. A polynomial d i=0 a i q i is palindromic if a i = a d i i. The motivation to study this problem comes from the topology of Schubert varieties. If W is crystallographic, then W is the Weyl group of some Kac-Moody group G and each element w W indexes a Schubert variety X w G/B. Topologically we have P w (q) = i 0 dim H i 2 (Xw ) q i. In [8], Carrell proves that the variety X w is rationally smooth if and only if P w (q) is palindromic. If W is a simply laced Weyl group, then rationally smooth is equivalent to smooth. To address this problem we define a weaker notion of palindromic. A polynomial d i=0 a i q i is k-palindromic if a i = a d i i k. Theorem 5. ([21]) Suppose m st 2 s, t S. For any w W, if P w (q) is 4-palindromic, then P w (q) is palindromic. Furthermore, suppose m st 2, 3 s, t S. Then every 2-palindromic P w (q) is palindromic. The theorem above states that, for many Coxeter groups, the palindromic property can be detected by looking at only a few coefficients of P w (q). A stronger version of Theorem 5 is given in [21, Theorem 1.2] in terms of triangle group avoidance. Theorem 5 is a consequence of the factorization theorem [21, Theorem 3.1]. In particular, we factor the polynomial P w (q) under the assumption that it is 2-palindromic. This factorization theorem yields several interesting enumeration results. For example, we compute an explicit formula for the generating series of the number of palindromic elements graded by length in the uniform Coxeter group W (m, n) (i.e. S = n and m st = m s, t S). 3 Schubert calculus for Kac-Moody groups In joint work with A. Berenstein from [3], we study the Schubert calculus of the flag variety G/B corresponding to a Kac-Moody group G. The structure of G is encoded by a generalized Cartan matrix (GCM), defined to be a square matrix A = (a i,j ) where a i,i = 2 and a i,j Z <0 if i j. Thus for each GCM, we can associate and study the cohomology ring H (G/B). Like the cohomology of the Grassmannian, H (G/B) has an additive basis of Schubert classes indexed by W, the Weyl group of G. We define the structure constants c w u,v by the product σ u σ v = w W c w u,v σ w. 3

4 In [3, Theorem 2.3], we give a formula for computing c w u,v in terms of the GCM A. This formula is based on the work of Kostant and Kumar in [14] where they define and study nil-hecke rings corresponding to Kac-Moody groups. While other formulas for Schubert structure constants exist (see [11]), it has been a long-standing open problem to find a formula that is combinatorially positive. Although it is well known from the geometry of G/B that the Schubert structure constants are non-negative integers, there are no known combinatorial proofs of this positivity (except in a few very special cases). Our new formula satisfies the following property. Theorem 6. ([3]) If the GCM A = (a i,j ) of G satisfies a i,j a j,i 4 (1) for all i, j, then the formula for c w u,v given in [3, Theorem 2.3] is combinatorially positive. In other words, the formula we construct is completely algebraic and the proof of positivity does not rely on the geometry of G/B. The condition (1) is precisely the condition that the Weyl group W has no braid relations or commuting relations as a Coxeter group. Theorem 6 above and [3, Theorem 2.3] have both been extended to include Schubert structure constants for the torus-equivariant cohomology HT (G/B) in [3]. 4 Recursive formulas for structure constants Let P Q be a pair of parabolic subgroups of a complex Lie group G and consider the induced sequence of partial flag varieties Q/P G/P G/Q. When comparing the three flag varieties above, the variety G/P typically has the most complicated cohomology structure. In [18, 19], I develop a recursive formula to compute Schubert structure coefficients of H (G/P ) in terms of the simpler cohomology rings H (Q/P ) and H (G/Q) under certain constraints. This formula is given in [19, Theorem 1.1]. One important class of coefficients satisfying these constraints of [19, Theorem 1.1] are coefficients c w u,v corresponding to Levi-movable triples (u, v, w) defined by Belkale and Kumar [2]. In [16], Ressayre shows that the set of Levi-movable triples, with c w u,v = 1, indexes the interior faces of the eigencone corresponding to the group G. By applying the recursive formula [19, Theorem 1.1] to Ressayre s work, I am able to determine the inclusion relations of the faces of the eigencone. In [17], N. Ressayre and I generalize the notion of Levi-movability to the setting of branching Schubert calculus. Branching Schubert calculus refers to the problem of computing the comorphism on cohomology rings induced from an equivariant embedding of one flag variety into another. If we consider the diagonal embedding of a flag variety into two copies of itself, then the comorphism on cohomology is simply the cup product. Hence, branching Schubert calculus is a generalization of usual Schubert calculus. We use the generalized definition of Levi-movable to give a more elegant solution to the branching eigenvalue problem. The main idea behind the proof of the recursive formula [19, Theorem 1.1] and its various applications to Levi-movability is to use the fact that Schubert structure coefficients count the number of points in the intersection of corresponding sets of Schubert varieties in general position. Since this intersection is transverse, we can apply tangent space analysis. 4

5 5 Projects currently in progress The following are some of my research projects currently in progress. It is likely that the results from Section 3 can be used to study torus-equivariant K-theory of Kac-Moody flag varieties. A. Berenstein and I are exploring this possibility and other generalizations. In [4], Bessenrodt, Luoto and van Willigenburg give a Littlewood-Richardson rule for noncommutative Schur functions that is a refinement of the classical Littlewood-Richardson rule for Schur functions. V. Tewari, S. van Willigenburg and I are looking for a geometric explanation of this refinement, analogous to how Schur functions are linked to the cohomology ring of the Grassmannian. The factorization theorem [21, Theorem 3.1] mentioned in Section 2 is, at the moment, combinatorial in nature. W. Slofstra and I are exploring a geometric explanation for this factorization. As an application, we hope to generalize the results of Ryan in [20] who proves that smooth Schubert varieties in the complete flag variety in type A are all towers of Grassmannian fibrations. References [1] D. Anderson, E. Richmond, and A. Yong. Eigenvalues of hermitian matrices and equivariant cohomology of grassmannians. Preprint. [2] P. Belkale and S. Kumar. Eigenvalue problem and a new product in cohomology of flag varieties. Invent. Math., 166(1): , [3] A. Berenstein and E. Richmond. Littlewood-richardson coefficients for reflection groups. Submitted. arxiv: [4] C. Bessenrodt, K. Luoto, and S. van Willigenburg. Skew quasisymmetric Schur functions and noncommutative Schur functions. Adv. Math., 226(5): , [5] B. G. Bodmann. Optimal linear transmission by loss-insensitive packet encoding. Appl. Comput. Harmon. Anal., 22(3): , [6] P. Boufounos, G. Kutyniok, and H. Rauhut. Sparse recovery from combined fusion frame measurements. IEEE Trans. Inform. Theory, to appear. [7] M. Bownik, K. Luoto, and E. Richmond. A combinatorial characterization of tight fusion frames. Submitted. arxiv: [8] J. B. Carrell. The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties. In Algebraic groups and their generalizations: classical methods (Uni. Park, PA, 1991), volume 56 of Proc. Sympos. Pure Math., pages AMS, Providence, RI, [9] P. G. Casazza, G. Kutyniok, and S. Li. Fusion frames and distributed processing. Appl. Comput. Harmon. Anal., 25(1): , [10] A. Chebira, M. Fickus, and D. G. Mixon. Filter bank fusion frames. preprint,

6 [11] H. Duan. Multiplicative rule of Schubert classes. Invent. Math., 159(2): , [12] A. A. Klyachko. Stable bundles, representation theory and Hermitian operators. Selecta Math. (N.S.), 4(3): , [13] A. Knutson and T. Tao. The honeycomb model of GL n (C) tensor products. I. Proof of the saturation conjecture. J. Amer. Math. Soc., 12(4): , [14] B. Kostant and S. Kumar. The nil Hecke ring and cohomology of G/P for a Kac-Moody group G. Adv. in Math., 62(3): , [15] G. Kutyniok, A. Pezeshki, R. Calderbank, and T. Liu. Robust dimension reduction, fusion frames, and Grassmannian packings. Appl. Comput. Harmon. Anal., 26(1):64 76, [16] N. Ressayre. Geometric invariant theory and the generalized eigenvalue problem. Invent. Math., 180(2): , [17] N. Ressayre and E. Richmond. Branching Schubert calculus and the Belkale-Kumar product on cohomology. Proc. Amer. Math. Soc., 139(3): , [18] E. Richmond. A partial Horn recursion in the cohomology of flag varieties. J. Algebraic Combin., 30(1):1 17, [19] E. Richmond. A multiplicative formula for structure constants in the cohomology of flag varieties. Michigan Math. J., 61(1):3 17, [20] K. M. Ryan. On Schubert varieties in the flag manifold of Sl(n, C). Math. Ann., 276(2): , [21] W. Slofstra and E. Richmond. Rationally smooth elements of coxeter groups and triangle group avoidance. Submitted. arxiv: