Identities Relating Schur s-Functions and Q-Functions by Elizabeth Angela DeWitt A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2012 Doctoral Committee: Professor John R. Stembridge, Chair Professor Andreas R. Blass Professor Sergey Fomin Associate Professor Kevin J. Compton Associate Professor Thomas Lam ACKNOWLEDGEMENTS First, Iwouldliketothankmyadvisor, ProfessorJohnStembridge, withoutwhom this work would not have been accomplished. From classes and reading to research and writing, he has been a patient teacher over the many years, interested in and helpful with every detail, guiding me to greater understanding. He has given me the freedom to explore while his suggestions and insights have called me back to the path of progress. I am thankful to have studied under his sound direction. I would like to thank my other committee members: to Sergey Fomin for intro- ducing me to symmetric functions in his class; to Andreas Blass for serving on my Preliminary Exam committee as well and for pointing out that when my parents explain my graduate work they can simply say that I counted to one or two; to Thomas Lam for his careful reading and good memory; and to Kevin Compton for his willingness to serve and his suggestions to improve the final draft. Thank you to the members of the Department of Mathematics at the University of Michigan. I am grateful that so many of my classes were taught by such great mathematicians and teachers. Thank you to Karen Rhea and the other introductory program faculty for their mentoring as I taught my students. Thank you to the department staff for their caring support, their encouragement, and their guidance through the University. Thank you to my fellow graduate students: there were many exhausting years, difficult classes, and late night homework and grading sessions I would not have survived without their company. ii Thank you to the faculty of Davidson College. I cannot imagine a better un- dergraduate experience, and I am grateful for the environment you created that encouraged and respected my individual voice. Thank you to the math faculty, and in particular Professor Benjamin Klein, who started me down the path to combina- torics. I now have original work to share at a long promised Math Coffee. I am blessed to have many friends. To my Davidson friends, thank you for those little chats as we struggled through graduate school and careers together though we live far apart. To my friends of Graduate Christian Fellowship, thank you for the prayerful encouragement, the meals when I needed to get over the first hurdles of writing, and the support at my defense. To my McCord Road Christian Church family, thank you for 25 years of prayers and faithfulness. Thank you to all of my family for your love and support. I am proud to be descended from two loving and intelligent grandmothers, Gertrude Rolfes and Geral- dineDeWitt, andrememberwithgreataffectionmygrandfathers, LouRolfesandDr. Paul DeWitt. Thank you to my brothers, Chris and Tim: they are my best friends for life and I am so blessed to have brothers who compensate for my weaknesses and call. Thank you to my parents, Kevin and Theresa, for their everlasting love, constant support, and faithful guidance (Isaiah 40:31) - I would not trade them. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v CHAPTER I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Partitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Diagrams and Tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Symmetric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 II. The Leading Term of Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 λ/µ 2.1 Unmarked Shifted Tableaux and the Leading Term of Q . . . . . . . . . . 12 λ/µ 2.2 Ascents and Descents of the Diagonal List of D(cid:48)(λ/µ) . . . . . . . . . . . . . 14 2.3 Proof of Theorem II.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 III. Schur s-Functions in ΩQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Domino Skimmings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Proof of Theorem III.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Proof of Lemma III.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 IV. Equality between S and Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 λ ν 4.1 The Shifted Littlewood-Richardson Rule . . . . . . . . . . . . . . . . . . . . 39 4.2 The Flip of D(cid:48)(λ/µ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3 Transformations leading to equal S-functions . . . . . . . . . . . . . . . . . . 43 4.4 Proof of Theorem IV.3: The Case µ=∅ . . . . . . . . . . . . . . . . . . . . 44 4.5 Proof of Theorem IV.3: The Case µ(cid:54)=∅ . . . . . . . . . . . . . . . . . . . . 52 4.6 Alternate Proof of Theorem IV.3: The Case mk/∅ . . . . . . . . . . . . . . . 56 V. Equality between s and Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 δ(n)/λ ν 5.1 The P-Expansion of s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 δ(n)/λ 5.2 Equality of s and P . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 δ(n)/λ ν∗ BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 iv LIST OF FIGURES Figure 2.1 The Four Shifted Diagrams D(cid:48)(λ/µ) with D(cid:48)(λ/µ)≥2 =D(cid:48)(765431/52) . . . . . . . 17 2.2 Free Entries in the Four Leading Term Tableau with D(cid:48)(λ/µ)≥2 =D(cid:48)(765431/52) . 21 3.1 Two Domino Skimmings of λ/µ=10,7,5,5/5,3,3 . . . . . . . . . . . . . . . . . . 30 3.2 The Cells That Cannot Be Included in a Domino Skimming . . . . . . . . . . . . . 33 3.3 Domino Skimmings of λ/µ=10,10,7,6,4,2,2/8,5,5,3 . . . . . . . . . . . . . . . 35 4.1 Complements and Flips of Shifted Diagrams . . . . . . . . . . . . . . . . . . . . . . 42 4.2 The (i,i) and (i+1,i+1) Hooks of Shifted Tableau T . . . . . . . . . . . . . . . 45 H 4.3 The Shifted Tableau of Unshifted Shape λ/µ=85/∅ . . . . . . . . . . . . . . . . . 47 4.4 Examples of Tableaux of Form T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 a 4.5 The (1,1)-, (2,2)-, and (3,3)-Hooks of a General Tableau of Form T . . . . . . . . 51 a 4.6 A Skew λ/µ with Constant Row and Column Lengths . . . . . . . . . . . . . . . . 55 5.1 An Example of Theorem V.3, s =P . . . . . . . . . . . . . . . . . . . 61 δ(9)/(53) H(53)∗ v CHAPTER I Introduction Schur s-functions and Q-functions are two important families within the alge- bra Λ of symmetric functions with applications for other fields. The (non-skew) s-functions form a basis for Λ and are indexed by partitions (i.e. weakly decreasing sequences of non-negative integers λ = (λ ≥ λ ≥ ···) converging to zero). They 1 2 are fundamental objects in several fields: for example, they are the characters of irreducible polynomial representations of GL (C), they encode the character tables n of the symmetric groups, and they represent Schubert classes in the cohomology ring of the Grassmann manifold of k-planes in n-space. The skew s-functions are indexed by pairs of partitions λ/µ such that µ ≤ λ for all i. They encode (not neces- i i sarily irreducible) representations that arise naturally, for example when branching from GL(m + n) to GL(m) × GL(n). Both non-skew and skew s-functions have a combinatorial description as generating functions for semistandard tableaux. Slightly less well known are the Q-functions. The (non-skew) Q-functions form a basis for a sub-algebra Ω of Λ and are indexed by strict partitions (i.e. partitions whose non-zero parts are strictly decreasing). They are also fundamental objects in several fields: for example, they are the characters of irreducible representations of a certain family of Lie superalgebras, they encode the character tables of the projective 1 2 representations of the symmetric groups, and they represent Schubert classes in the cohomology ring of the isotropic Grassmannian in n-space. The skew Q-functions are indexed by pairs of strict partitions, and both non-skew and skew Q-functions are expressible as generating functions over shifted tableaux. One of the stranger recent observations about skew s-functions is the discovery that there are unexpected collisions among them; that is, there are equalities s = λ/µ s with no combinatorially or algebraically obvious explanation , with the smallest α/β such collision occurring in degree 8. The question of when two (possibly skew) s- functions are equal has been studied in a series of papers by van Willigenburg ([13]), Billera, Thomas and van Willigenburg ([3]), Reiner, Shaw, and van Willigenburg ([9]), and McNamara and van Willigenburg ([7]), but so far the results are not definitive and the complete answer appears to be complicated. McNamara and van Willigenburg([7])unifytheworktodate, providingasufficiencyconditionexplaining allcurrentlyknowncollisionsandconjecturingarelatednecessaryconditionthatmay provide an algorithm for generating all s-function collisions. Since Q-functions play a similar role in the sub-algebra Ω to the role of s-functions in Λ, and since both families have a combinatorial description in terms of tableux, it is natural to ask similar questions of Q-functions and the elements of Ω to those that are posed of s-functions within Λ. Whereas s-functions are monic, Q-functions are not; in determining equality up to constant multiple between possibly skew Q- functions, we begin in Chapter 2 by finding the leading coefficient of any skew Q- function (Theorem II.6). In Chapter 3 we turn to the question of equality up to constant multiple between an s-function and a Q-function, reducing the s-functions we need to consider by finding all the s-functions in Ω (Theorem III.5). In Chapter 4 we explore equality among Q-functions, and in particular, determine all the S- 3 functions that are a constant multiple of a single non-skew Q-function (Theorem IV.3). We return in Chapter 5 to relations between s- and Q-functions, providing a Q-expansion for all s-functions in Ω with non-negative coefficients (Theorem V.5) and determining all the s-functions that are a constant multiple of a single non-skew Q-function (Theorem V.3). 1.1 Partitions Definition I.1. A partition λ is a sequence (λ ,λ ,...,λ ,...) of weakly decreasing 1 2 i non-negative integers containing finitely many non-zero terms. We will consider two partitions equivalent if they differ only by the number of zero terms. Definition I.2. The ith part of partition λ, λ , is the ith largest term of λ. i Definition I.3. The length of λ, l(λ), is the number of non-zero parts of λ. Definition I.4. The size of λ, |λ|, is the sum of the parts of λ. Definition I.5. A strict partition is a partition such that all non-zero parts are distinct, i.e. there does not exist an i such that λ = λ (cid:54)= 0. i i+1 Definition I.6. The k-staircase partition, δ(k), is the partition (k,k −1,...,2,1). For example, δ(5) is the partition 54321. Definition I.7. On the set of partitions of size n, the lexicographic order is a total order such that, for partitions λ and µ, λ ≥ µ if either λ = µ or the first non-zero difference λ −µ is positive. i i Definition I.8. On the set of partitions of size n, the dominance order is a partial order such that, for partitions λ and µ, λ ≥ µ if λ +···+λ ≥ µ +···+µ for each 1 i 1 i i ≥ 1. 4 1.2 Diagrams and Tableaux Definition I.9. For any partition λ, we define the (unshifted) diagram D(λ) to be {(i,j) ∈ Z2 : 1 ≤ j ≤ λ ,i ≥ 1}. As with matrices, we will use the convention that i the (i,j) position of D(λ) is the cell in the ith row from the top and the jth column from the left. Notation I.10. We will use µ ⊆ λ to represent that the diagram of µ is contained in the diagram of λ, i.e. µ ≤ λ for all i ≤ l(µ). i i Definition I.11. For any partitions λ and µ with µ ⊆ λ, we define the skew (un- shifted)diagramD(λ/µ)tobeD(λ)\D(µ)assets,i.e.{(i,j) ∈ Z2 : µ < j ≤ λ ,i ≥ 1}. i i Definition I.12. For a partition λ, we define the conjugate of λ, λ(cid:48), to be the partition whose diagram D(λ(cid:48)) interchanges the rows and columns of D(λ); thus λ(cid:48) i is the number of cells in the ith column of D(λ). In the skew case, we will use λ(cid:48)/µ(cid:48) to denote the conjugate of λ/µ, where D(λ(cid:48)/µ(cid:48)) = D(λ(cid:48))\D(µ(cid:48)). Definition I.13. For any strict partition λ, we define the shifted diagram D(cid:48)(λ) to be {(i,j) ∈ Z2 : i ≤ j ≤ λ +i−1,i ≥ 1}, so that D(cid:48)(λ) may be viewed as the i diagram formed when the ith row of the unshifted diagram D(λ) is shifted (i−1) positions to the right. As with matrices, we will use the convention that the (i,j) position of D(cid:48)(λ) is the cell in the ith row from the top and the jth column from the left. Since the rows are not left-justified and are instead a staircase-like shape, for all D(cid:48)(λ) if i > j then (i,j) ∈/ D(cid:48)(λ). Definition I.14. For any strict partitions λ and µ with µ ⊆ λ, we define the skew shifted diagram D(cid:48)(λ/µ) to be D(cid:48)(λ)\D(cid:48)(µ) as sets, i.e. {(i,j) ∈ Z2 : µ +i−1 < i j ≤ λ +i−1,i ≥ 1}. i 5 Definition I.15. The kth diagonal of D(cid:48)(λ/µ), counting from left to right, is the set {(i,k+i−1) ∈ D(cid:48)(λ/µ)}. ViewingD(cid:48)(λ/µ)asD(λ/µ)withtheithrowshifted(i−1) positions to the right, the kth diagonal of D(cid:48)(λ/µ) is the image of the kth column of D(λ/µ), and so the length of (or number of cells in) the kth diagonal is λ(cid:48) −µ(cid:48). k k In particular, the main or first diagonal of D(cid:48)(λ/µ) is the set {(i,i) ∈ D(cid:48)(λ/µ)} and has length l(λ)−l(µ). Definition I.16. We define any skew unshifted (or shifted) diagram D(λ/µ) (or D(cid:48)(λ/µ)) to be connected if one can move between any two cells in D(λ/µ) (or D(cid:48)(λ/µ)) through a series of horizontal and vertical steps while remaining within the diagram. Definition I.17. For any partitions λ and µ with µ ⊆ λ, an (unshifted) tableau of shape λ/µ is an assignment to the positions in D(λ/µ) of symbols from the ordered alphabet P = {1 < 2 < 3 < ···} such that the entries are weakly increasing left to right across each row right and are strictly increasing down each column. Definition I.18. The content of (unshifted) tableau T is the sequence (α ,α ,...) 1 2 where α is the number of a entries in T for each a ≥ 1. a Definition I.19. We define two unshifted shapes to be combinatorially equivalent if there exists a bijection between the positions such that each tableau of one shape corresponds to a valid tableau of the other. For example, two shapes are combi- natorially equivalent if they differ by the insertion or deletion of a column or row of length zero, by the translation of connected components, by the reordering of connected components, or by some combination of these. Notation I.20. We will let P(cid:48) be the alphabet of ordered symbols {1(cid:48) < 1 < 2(cid:48) < 2 < ···}, where the entries 1,2,... are said to be unmarked and the entries 1(cid:48),2(cid:48),...