Combinatorial constructions motivated by K-theory of Grassmannians A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Rebecca Patrias IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy Pavlo Pylyavskyy July, 2016 c Rebecca Patrias 2016 ALL RIGHTS RESERVED Acknowledgements I would like to first thank my advisor, Pasha Pylyavskyy, for his guidance and support. I learned a lot from him and enjoyed our time working together. I thank Vic Reiner for always having his door open and being a great mentor. I thank him for encouraging me to present Pasha and Thomas Lam's paper on K-theoretic combinatorial Hopf algebras in his topics course; this reading led to Chapter 3 of this thesis. I am grateful to Gregg Musiker, Dennis Stanton, and Peter Webb for introducing me to new kinds mathematics and for sharing their curiosity and excitement. I thank the current and former Minnesota combinatorics graduate students Alex Csar, Elise DelMas, Kevin Dilks, Theo Douvropoulos, Thomas McConville, Emily Gu- nawan, Alex Rossi Miller, and Nathan Williams for giving advice and encouragement freely. I am deeply indebted to my professors at Carleton College, particularly Mark Kruse- meyer, Eric Egge, Helen Wong, and Gail Nelson for being amazing instructors, role models, and friends. I would not have made it to|let alone through|graduate school without the sup- port and guidance of my SMP family. I could easily list dozens of people, but I'll limit myself to mentioning Deanna Haunsperger, Steve Kennedy, Pam Richardson, Margaret Robinson, Erica Flapan, Liz Stanhope, Becky Swanson, Amanda Croll, Emily Olson, and Christina Knudson. I cannot possibly express how important their support has been. I am grateful to my current officemates, Cora Brown, Sunita Chepuri, Harini Chan- dramouli, and Dario Valdebenito, for making our office the envy of grad students ev- erywhere. I appreciate the conversation, distraction, snacks, and gossip. i I thank JJ Abrahms, Gene Rodenberry, and Shonda Rhimes for entertaining me and for preparing me for the emotional ups and downs of graduate school by continuing to kill off or otherwise get rid of my favorite characters and then sometimes bringing them back. Lastly, I would like to thank Alexander Garver and my friends and family for their continued love and support. ii Dedication In memory of John Patrias iii Abstract Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and the Malvenuto- Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map in these cases. Next, using the Hecke insertion of Buch-Kresch-Shimozono-Tamvakis-Yong and the K-Knuth equivalence of Buch-Samuel in place of Robinson-Schensted and Knuth equiv- alence, we introduce a K-theoretic analogue of the Poirier-Reutenauer Hopf algebra of standard Young tableaux. As an application, we rederive the K-theoretic Littlewood- Richardson rules of Thomas-Yong and of Buch-Samuel. Lastly, we define a K-theoretic analogue of Fomin's dual graded graphs, which we call dual filtered graphs. The key formula in this definition is DU −UD = D+I. Our major examples are K-theoretic analogues of Young's lattice, of shifted Young's lattice, and of the Young-Fibonacci lattice. We suggest notions of tableaux, insertion algorithms, and growth rules whenever such objects are not already present in the literature. We also provide a large number of other examples. Most of our examples arise via two constructions, which we call the Pieri construction and the M¨obiusconstruction. The Pieri construction is closely related to the construction of dual graded graphs from a graded Hopf algebra as described by Bergeron-Lam-Li, Lam-Shimozono, and Nzeutchap. The M¨obiusconstructions is more mysterious but also potentially more important as it corresponds to natural insertion algorithms. iv Contents Acknowledgements i Dedication iii Abstract iv List of Figures x 1 Introduction 1 1.1 Schur functions and cohomology of the Grassmannian . .1 1.2 Stable Grothendieck polynomials and K-theory of the Grassmannian . .3 1.3 Overview . .4 1.4 Summary of the main results . .5 2 Preliminaries 7 2.0.1 Partitions, tableaux, and Schur functions . .7 2.0.2 Compositions . .8 2.0.3 The Robsinson-Schensted-Knuth correspondence . .9 3 Combinatorial Hopf algebras and antipode maps 11 3.1 Introduction . 11 3.1.1 Chapter overview . 12 3.2 Hopf algebra basics . 13 3.2.1 Algebras and coalgebras . 13 3.2.2 Morphisms and bialgebras . 15 v 3.2.3 The antipode map . 16 3.3 The Hopf algebra of quasisymmetric functions . 19 3.3.1 Monomial quasisymmetric functions . 19 3.3.2 (P; θ)-partitions . 20 3.3.3 Fundamental quasisymmetric functions . 21 3.3.4 Hopf structure . 23 3.4 The Hopf algebra of multi-quasisymmetric functions . 25 3.4.1 (P; θ)-set-valued partitions . 25 3.4.2 The multi-fundamental quasisymmetric functions . 26 3.4.3 Hopf structure . 29 3.5 The Hopf algebra of noncommutative symmetric functions . 31 3.5.1 Noncommutative ribbon functions . 31 3.5.2 Hopf structure . 31 3.6 The Hopf algebra of Multi-noncommutative symmetric functions . 32 3.6.1 Multi-noncommutative ribbon functions and bialgebra strucure . 33 3.6.2 Antipode map for MNSym . 34 3.6.3 Antipode map for mQSym . 38 3.7 A new basis for mQSym . 38 3.7.1 (P; θ)-multiset-valued partitions . 38 3.7.2 Properties . 39 3.7.3 Antipode . 43 3.8 The Hopf algebra of symmetric functions . 43 3.9 The Hopf algebra of multi-symmetric functions . 44 3.9.1 Set-valued tableaux . 44 3.9.2 Basis of stable Grothendieck polynomials . 45 3.9.3 Weak set-valued tableaux . 45 3.10 The Hopf algebra of Multi-symmetric functions . 46 3.10.1 Reverse plane partitions . 46 3.10.2 Valued-set tableaux . 47 3.11 Antipode results for mSym and MSym . 48 vi 4 K-theoretic Poirier-Reutenauer bialgebra 55 4.1 Introduction . 55 4.1.1 K-Poirier-Reutenauer bialgebra and Littlewood-Richardson rule 56 4.1.2 Chapter overview . 59 4.2 Poirier-Reutenauer Hopf algebra . 60 4.2.1 Two versions of the Littlewood-Richardson rule . 61 4.3 Hecke insertion and the K-Knuth monoid . 63 4.3.1 Hecke insertion . 63 4.3.2 Recording tableaux . 67 4.3.3 K-Knuth equivalence . 69 4.3.4 Properties of K-Knuth equivalence . 70 4.4 K-theoretic Poirier-Reutenauer . 71 4.4.1 K-Knuth equivalence of tableaux . 71 4.4.2 Product structure . 72 4.4.3 Coproduct structure . 74 4.4.4 Compatibility and antipode . 76 4.5 Unique Rectification Targets . 77 4.5.1 Definition and examples . 78 4.5.2 Product and coproduct of unique rectification classes . 79 4.6 Connection to symmetric functions . 80 4.6.1 Symmetric functions and stable Grothendieck polynomials . 80 4.6.2 Weak set-valued tableaux . 83 4.6.3 Fundamental quasisymmetric functions revisited . 85 4.6.4 Decomposition into fundamental quasisymmetric functions . 86 4.6.5 Map from the KPR to symmetric functions . 88 4.7 Littlewood-Richardson rule . 89 4.7.1 LR rule for Grothendieck polynomials . 89 4.7.2 Dual LR rule for Grothendieck polynomials . 91 5 Dual filtered graphs 93 5.1 Introduction . 93 5.1.1 Weyl algebra and its deformations . 93 vii 5.1.2 Differential vs difference operators . 95 5.1.3 Pieri and M¨obiusconstructions . 96 5.1.4 Chapter overview . 97 5.2 Dual graded graphs . 98 5.2.1 Dual graded graphs . 98 5.2.2 Growth rules as generalized RSK . 102 5.3 Dual filtered graphs . 105 5.3.1 The definition . 105 5.3.2 Trivial construction . 105 5.3.3 Pieri construction . 106 5.3.4 M¨obiusconstruction . 107 5.4 Major examples: Young's lattice . 108 5.4.1 Pieri deformations of Young's lattice . 108 5.4.2 M¨obiusdeformation of Young's lattice . 110 5.4.3 Hecke growth and decay . 112 5.4.4 M¨obiusvia Pieri . 118 5.5 Major examples: shifted Young's lattice . 120 5.5.1 Pieri deformation of shifted Young's lattice . 120 5.5.2 M¨obiusdeformation of shifted Young's lattice . 123 5.5.3 Shifted Hecke insertion . 125 5.5.4 Shifted Hecke growth and decay . 133 5.6 Major examples: Young-Fibonacci lattice . 141 5.6.1 M¨obiusdeformation of the Young-Fibonacci lattice . 141 5.6.2 K-Young-Fibonacci insertion . 143 5.6.3 KYF growth and decay . 149 5.7 Other examples . 154 5.7.1 Binary tree deformations . 154 5.7.2 Poirer-Reutenauer deformations . 158 5.7.3 Malvenuto-Reutenauer deformations . 160 5.7.4 Stand-alone examples . 162 5.8 Enumerative theorems via up-down calculus . 164 5.9 Affine Grassmannian . 166 viii 5.9.1 Dual k-Schur functions . 167 5.9.2 Affine stable Grothendieck polymonials . 168 5.9.3 Dual graded and dual filtered graphs . 170 References 174 ix List of Figures 3.1 Diagram of combinatorial Hopf algebras . 11 3.2 Diagram of K-theoretic combinatorial Hopf algebras of Lam and Pylyavskyy 12 3.3 Multiplying R(2;2) and R(1;2) ......................... 32 ~ ~ 3.4 Multiplying R(2;2) and R(1;2) ........................