Research Statement. Nicholas Teff:

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1 Research Statement Nicholas Teff: () The symmetric group S n is everywhere in mathematics. A fundamental example of this is the representation theory of S n. A representation is simply a vector space invariant under a linear action of S n. Important examples include the irreducible representations which are the building blocks of all other representations. By the early 20 th century Alfred Young constructed the irreducible representations using the combinatorics of partitions and tableaux. Seventy years later T.A. Springer discovered that the cohomology ring of geometric spaces, now called Springer varieties, also contain the irreducible representations. The connections between combinatorics and the geometry of Springer varieties is still studied today. I use the combinatorics of S n to construct new representations called Hessenberg representations. These representations are interesting for several reasons. First, much like the Springer representation they are cohomology rings of algebraic varieties, therefore of interest to both geometers and representation theorists. Second, I employ a combinatorial construction called GKM theory [GKM98] which effectively reduces typically difficult computations to polynomial multiplication. Third, the combinatorial nature of the construction exposes new connections between the combinatorics, representation theory, and geometry. Fundamentally, I seek to determine the irreducible components of Hessenberg representations. This work is a generalization of (i) Springer representations which geometrically constructs the irreducible representations S λ of S n [DCP8, Spr76]; (ii) Schubert calculus which uses a S n representation to define a linear basis of the cohomology of the flag variety [Bil99, KK86, Tym08]; and (iii) a S n representation on the ordinary cohomology of the Coxeter complex which can be used to prove the symmetry and unimodality of a refinement of the Eulerian numbers [Pro90, Ste92]. Several goals of this research include; () Determine the structure of Hessenberg representations. Evidence suggests Hessenberg representations have additional structure as permutation representations. I have proven this for parabolic Hessenberg representations [Tef]. (2) Create a generalized Schubert calculus. Classical Schubert calculus use a representation of S n to construct Schubert classes in the equivariant cohomology of the flag variety [Bil99, KK86, Tym08]. Employing the Hessenberg representation, I have successfully generalized the classical approach and construct a linear basis for the highest root Hessenberg representation [Tefa]. (3) Prove the Shareshian-Wachs conjecture for chromatic symmetric functions. Shareshian and Wachs [SW] have conjectured that certain chromatic symmetric functions are the Frobenius characteristic of Hessenberg representations. I have proven this conjecture in the case of parabolic and highest root Hessenberg representations [Tefb]. For simplicity in this research statement I discuss S n, but much of this research is true for a class of groups called Coxeter groups. Understanding the combinatorics of Coxeter groups are primary research goals of mine. In addition to Hessenberg representations I have ongoing projects studying Iwahori-Hecke algebras; pattern avoidance for Coxeter groups; and ideals in the root lattice of a finite Coxeter group. Furthermore, I want to explore more exciting connections between algebraic combinatorics and combinatorial representation theory.

2 The Hessenberg Representation Here we define Hessenberg representations. The simple transpositions s i = (i i + ) generate S n. For w S n define l(w) to be the minimal length of any expression w = s i s i2 s ik. The Bruhat graph has vertices S n and u, w S n are connected by an edge u w if l(w) > l(u) and w = (ij)u. A Hessenberg function is a non-decreasing function h : {, 2,, n} {, 2,, n} which also satsifies h(i) i. The GKM graph of h is the subgraph of the Bruhat graph with vertices S n and edges u w when w (i) h(w (j)). h = 23 h = 223 h = 233 h = 333 Bruhat graph Figure. GKM graphs The GKM ring of h is a subring of Maps(S n, C[t, t 2,, t n ]) defined from the edges of the GKM graph { } HT for each edge w (ij)w (h) := P : S n C[t,, t n ] :. P(w) P((ij)w) t i t j We will also be interested in a finite dimensional scalar restriction H (h) := H T (h) C[t,,t n] C. Geometrically, this is the GKM presentation of the (equivariant and ordinary) cohomology of a family of algebraic varieties called Hessenberg varieties [DMPS92, GKM98]. The GKM approach is advantageous because it gives new and interesting combinatorially defined representations, and effectively reduces computations in cohomology to polynomial multiplication. A small survey of recent work employing the GKM presentation include [GT09, GZ03, KT03, Tym08]. Finally, we need a linear action of S n. If u, w S n and P HT (h), then (w P)(u) = w(p(w u)), where w is acting on the polynomial P(w u) by permuting variables. This action is well defined on H (h), so we have two representations to investigate. Question. Given h, what is irreducible representation decomposition of H T (h) or H (h)? Before I began explicit decompositions of H (h) were known in only three cases (i) when h(i) = i the graph is just the vertices S n on which S n acts by left multiplication, i.e. the left regular representation; (ii) when h(i) = n the graph is the Bruhat graph and the representation is known to be trivial [Tym08]; and (iii) when h(i) = i + the graph is the permutohedron of order n. The irreducible decomposition is more complicated but was first obtained by Procesi [Pro90]. 2

3 Parabolic Hessenberg fucntions. My first result provides a partial solution to Question. In fact, it provides a picture of the evolution of the Hessenberg representation from the left regular representation to the trivial representation. Fix a sequence λ = (λ λ 2 λ n ) Z n 0 such that λ λ j = n (i.e. a partition). A parabolic Hessenberg function is one defined as h(i ) = λ for i λ, h(i 2 ) = λ + λ 2 for λ + i 2 λ + λ 2 etc... These functions have disconnected GKM graphs where each component is isomorphic to the Bruhat graph of the Young subgroup S λ := S λ S λn. For each λ the permutation representation M λ is obtained from the action of S n on the set of (left) cosets of S λ. Theorem ([Tef]). Let h be the parabolic Hessenberg function defined by λ. Then the Hessenberg representation on H (h λ ) = S λ M λ. This subsumes the results mentioned above for h(i) = i where λ = (,,, ) and h(i) = n where λ = (n). Additionally, this is a coarse decomposition of the Hessenberg representation. It also proves many of the representations have additional structure as permutation representations. Since, every representation is a direct sum of irreducible representations S λ it is therefore stronger to be able to express a representation as a direct sum of the permutation representations M λ. Project. Prove H (h) is always a direct sum of permutation representations. One way to prove this is to exhibit a basis of H (h) which is permuted by S n such that each element of the the basis is stabilized by some Young subgroup. The next section describes my progress towards constructing ANY basis let alone a permutation basis. An interesting consequence of Theorem is an upper triangular decomposition of the representation. A classical result called Young s rule states the irreducible decomposition of M λ is () M λ = partitions µ λ K λµ S µ where the multiplicity K λµ is the well-known Kostka matrix which is upper triangular with respect to the lexicographic order, i.e. µ λ if the first non-vanishing of µ i λ i is positive. Define a partial order on Hessenberg functions by taking the transitive closure of the relation h < h if and only if there exists an i {, 2,, n} such that h (i) = h(i) + and h (j) = h(j) for all j i. Theorem 2 ([Tefb]). Let h λ and h µ be the functions defined by λ and µ as above, then h µ > h λ if and only if µ > λ. Therefore, the irreducible representation S µ is a direct summand of H (h λ ) if and only if h µ > h λ. Computer calculations for n 6 verify a weakened version of this theorem. Project 2. Prove that if h > h and S λ is a direct summand of H (h), then S λ is a direct summand of H (h ). In another direction I have studied the S n invariants, i.e. P H (h) such that w P = P for all w S n. Determining the dimension of this space determines the multiplicity of the trivial irreducible representation S (n). Project 3. Determine a basis of the space of invaraints. I have a conjectured basis for this based on paths deleted from the Bruhat graph. This project generalizes to any finite reflection group, and it truly a problem about the lower order ideals of any root system. 3

4 Generalized Schubert Calculus A classical problem of Schubert calculus is to define explicit cohomology classes S w to represent the linear basis of Schubert classes for the flag variety. In the GKM presentation this reduces to finding polynomials S u (w) which are nonzero when there is a directed path from w to v in the Bruhat graph. For general applications of GKM rings existence of generalized Schubert classes called flow-up classes is not guaranteed [GT09, GZ03]. Fortunately, BRIEF for ARTICLE the Hessenberg representation they exist but are not unique. THE AUTHOR 0 [ss2s] t t 2 [ss2s] t 3 t 2 [ss2] [s2s] 0 t t 2 [ss2] [s2s] 0 t t 2 [s] [s2] 0 t t 2 [s] [s2] 0 [e] 0 [e] 0 Figure 2. Non-unique flow-classes h = {t 2 t } h = h =Φ [ss2s] [ss2s] [ss2s] Project 4. Determine = t t 2 a combinatorial [ss2] [s2s] algorithm [ss2] for flow-up [s2s] classes [ss2] in H [s2s] T (h)? Classically, the classes = t 2 t 3 S w are constructed by divided difference operators [KK86, Tym08]. These are recursive maps which = t t 3 define S siw if S w is known and l(s [s] [s2] [s] [s2] i w) < l(w). They are excellent tools [s] [s2] for inductive arguments. For example, Billey used the divided difference operator to obtain a closed [e] [e] [e] formula for S w (v) in terms of minimal length expressions for w, v [Bil99]. Using GKM theory divided difference operators were newly constructed from the S n representation [Tym08]. Project 5. Define divided difference operators for all H T (h)? I have made a significant first step along these lines. h() = n and h(i) = n otherwise. The highest root Hessenberg function is Theorem 3 ([Tefa]). Let h the highest root Hessenberg function. There exist divided difference operators i : HT (h) H T (h) and flow-up classes {Pw } w Sn such that { i P w P s iw if l(s i w) < l(w); = 0 if l(s i w) > l(w). Further, if w = s i s in is a minimal length expression, then w := i in is well-defined. The result proves two things simultaneously (i) it generalizes the classical divided difference operator for the flag variety; and (ii) it gives matrix representations for the action of s i on the basis of flow-up classes. Generalizing this result is important, because without a basis it is difficult to analyze the structure of the Hessenberg representation. Project 6. Relate the flow-up classes P w to the Schubert classes S w. Billey s formula [Bil99] and a solution to Project 6 would give a closed combinatorial formula for the flow-up classes. This would be a great tool for geometers who want to obtain closed formulas for the products of Schubert classes, or more general flow-up classes. 4

5 Chromatic symmetric functions Let G be a graph with vertices V = {, 2,, n}. Stanley defined the chromatic symmetric function of G as X G (x, x 2, ) := x κ() x κ(n) κ where the sum is over proper vertex-colorings κ of G [Sta95]. For each function h there is graph which arises as the incomparability graph of a poset called a semi-order h = 23 h = 223 h = 233 h = 333 Figure 3. Incomparbility graphs. In 992 Stanley and Stembdrige conjectured the h-chromatic symmetric function X h (x, x 2, ) has positive integer coefficients when expanded in the basis of elementary symmetric functions [Sta95]. Shareshian and Wachs recently define a q-refinement of X h (x, x 2, ) [SW]. They prove that certain Eulerian polynomials appear in a specialization of the q-refined X h (x, x 2, ). The coefficients of these polynomials are the Betti numbers of H (h) [DMPS92]. Even more appears true. The Frobenius characteristic ch is an isometry between the representation ring of S n and the ring of symmetric functions Λ. The Shareshian-Wachs conjecture states Question 2 (Shareshian-Wachs [SW]). Let H (h) be the Hesesnberg representation and X h (x, x 2, ) the chromatic symmetric function. Are they related by the rule ch(h (h)) = ωx h (x, x 2, )? This conjecture remains open, but I have proven a large family of special cases. Theorem 4 ([Tefb]). The Shareshian-Wachs conjecture is true when h is either a parabolic or the highest root Hessenberg function. I am working on an interesting combinatorial approach to this problem Figure 4. Row standard h-tableaux. Project 7. Give a bijection between row standard h-tableaux and pairs of semi-standard tableaux and standard h-tableaux. Standard h-tableaux count the multiplicity of irreducible representation S λ in ch (ωx h (x, x 2, ). Solutions to this project generalize the RSK correspondence [Cho99, SWW97] and most importantly prove Question 2. For example, I have proven another large family of special cases. Theorem 5. Suppose h avoids the pattern (2344) then the Shareshian-Wachs conjecture is true. The chromatic symmetric function is an exciting application of Hessenberg representations. Not only does it reveal that the combinatorics of the symmetric group is deeply influences the geometry of Hessenberg varieties, but it weaves the representation theory together as well. 5

6 References [Bil99] Sara C. Billey, Kostant polynomials and the cohomology ring for G/B, Duke Math. J. 96 (999), no., MR (2000a:4060) [Cho99] Timothy Y. Chow, Descents, quasi-symmetric functions, Robinson-Schensted for posets, and the chromatic symmetric function, J. Algebraic Combin. 0 (999), no. 3, MR (2000m:05229) [DCP8] Corrado De Concini and Claudio Procesi, Symmetric functions, conjugacy classes and the flag variety, Invent. Math. 64 (98), no. 2, MR (82m:4030) [DMPS92] F. De Mari, C. Procesi, and M. A. Shayman, Hessenberg varieties, Trans. Amer. Math. Soc. 332 (992), no. 2, MR (92j:4060) [GKM98] Mark Goresky, Robert Kottwitz, and Robert MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 3 (998), no., MR (99c:55009) [GT09] Rebecca F. Goldin and Susan Tolman, Towards generalizing Schubert calculus in the symplectic category, J. Symplectic Geom. 7 (2009), no. 4, MR (200k:5343) [GZ03] Victor Guillemin and Catalin Zara, The existence of generating families for the cohomology ring of a graph, Adv. Math. 74 (2003), no., MR (2004f:53099) [KK86] Bertram Kostant and Shrawan Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G, Adv. in Math. 62 (986), no. 3, MR (88b:7025b) [KT03] Allen Knutson and Terence Tao, Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J. 9 (2003), no. 2, MR (2006a:4088) [Pro90] C. Procesi, The toric variety associated to Weyl chambers, Mots, Lang. Raison. Calc., Hermès, Paris, 990, pp MR (94k:4045) [Spr76] T. A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (976), MR (56 #49) [Sta95] Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Adv. Math. (995), no., MR (96b:0574) [Ste92] John R. Stembridge, Eulerian numbers, tableaux, and the Betti numbers of a toric variety, Discrete Math. 99 (992), no. -3, MR (93f:0503) [SW] J. Shareshian and M. L. Wachs, Chromatic quasisymmetric functions and Hessenberg varieties, ArXiv e- prints (20). [SWW97] Thomas S. Sundquist, David G. Wagner, and Julian West, A Robinson-Schensted algorithm for a class of partial orders, J. Combin. Theory Ser. A 79 (997), no., MR (98c:0562) [Tefa] Nicholas Teff, Divided difference operator for the highest root Hessenberg variety., In preparation. [Tefb] [Tef], Parabolic Hessenberg representations and chromatic symmetric functions, In preparation., Representations on Hessenberg varieties and Young s rule, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 20), Discrete Math. Theor. Comput. Sci. Proc., AO, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 20, pp MR (202j:403) [Tym08] Julianna S. Tymoczko, Permutation representations on Schubert varieties, Amer. J. Math. 30 (2008), no. 5, MR (2009h:4088) 6