Lecture Notes on Algebraic Combinatorics Jeremy L. Martin [email protected] December 3, 2012 Copyright c 2012 by Jeremy L. Martin. These notes are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 2 Foreword The starting point for these lecture notes was my notes from Vic Reiner's Algebraic Combinatorics course at the University of Minnesota in Fall 2003. I currently use them for graduate courses at the University of Kansas. They will always be a work in progress. Please use them and share them freely for any research purpose. I have added and subtracted some material from Vic's course to suit my tastes, but any mistakes are my own; if you find one, please contact me at [email protected] so I can fix it. Thanks to those who have suggested additions and pointed out errors, including but not limited to: Logan Godkin, Alex Lazar, Nick Packauskas, Billy Sanders, Tony Se. 1. Posets and Lattices 1.1. Posets. Definition 1.1. A partially ordered set or poset is a set P equipped with a relation ≤ that is reflexive, antisymmetric, and transitive. That is, for all x; y; z 2 P : (1) x ≤ x (reflexivity). (2) If x ≤ y and y ≤ x, then x = y (antisymmetry). (3) If x ≤ y and y ≤ z, then x ≤ z (transitivity). We'll usually assume that P is finite. Example 1.2 (Boolean algebras). Let [n] = f1; 2; : : : ; ng (a standard piece of notation in combinatorics) and let Bn be the power set of [n]. We can partially order Bn by writing S ≤ T if S ⊆ T . 123 123 12 12 13 23 12 13 23 1 2 1 2 3 1 2 3 The first two pictures are Hasse diagrams. They don't include all relations, just the covering relations, which are enough to generate all the relations in the poset. (As you can see on the right, including all the relations would make the diagram unnecessarily complicated.) Definition 1.3. Let P be a poset and x; y 2 P . • x is covered by y, written x l y, if x < y and there exists no z such that x < z < y. • The interval from x to y is [x; y] := fz 2 P j x ≤ z ≤ yg: (This is nonempty if and only if x ≤ y, and it is a singleton set if and only if x = y.) 3 The Boolean algebra Bn has a unique minimum element (namely ;) and a unique maximum element (namely [n]). Not every poset has to have such elements, but if a poset does, we'll call them 0^ and 1^ respectively (or if necessary 0^P and 1^P ). Definition 1.4. A poset that has both a 0^ and a 1^ is called bounded.1 An element that covers 0^ is called an atom, and an element that is covered by 1^ is called a coatom. (For example, the atoms in Bn are the singleton subsets of [n].) We can make a poset P bounded: define a new poset P^ by adjoining new elements 0^; 1^ such that 0^ < x < 1^ for every x 2 P . Meanwhile, sometimes we have a bounded poset and want to delete the bottom and top elements. Definition 1.5. A subset C ⊂ P is called a chain if its elements are pairwise comparable. Thus every chain is of the form C = fx0; : : : ; xng, where x0 < ··· < xn. An antichain is a subset of P in which no two of its elements are comparable.2 123 123 123 123 12 13 23 12 13 23 12 13 23 12 13 23 1 2 3 1 2 3 1 2 3 1 2 3 φ φ φ φ chain antichain antichain neither 1.2. Ranked Posets. One of the many nice properties of Bn is that its elements fall nicely into horizontal slices (sorted by their cardinalities). Whenever S l T , it is the case that jT j = jSj + 1. A poset for which we can do this is called a ranked poset. However, it would be tautological to define a ranked poset to be a poset in which we can rank the elements! The actual definition of rankedness is a little more subtle, but makes perfect sense after a little thought. 3 Definition 1.6. A chain x0 < ··· < xn is saturated if it is not properly contained in any other chain from x0 to xn; equivalently, if xi−1 l xi for every i 2 [n]. In this case, the number n is the length of the chain A poset P is ranked if for every x 2 P , all saturated chains with top element x have the same length; this number is called the rank of x and denoted r(x). It follows that (1.1) x l y =) r(y) = r(x) + 1: A poset is graded if it is ranked and bounded. Note: (1) \Length" means the number of steps, not the number of elements | i.e., edges rather than vertices in the Hasse diagram. 1This has nothing to do with the more typical metric-space definition of \bounded". 2To set theorists, \antichain" means something stronger: a set of elements such that no two have a common lower bound. This concept does not typically arise in combinatorics, where many posets are bounded. 3Sometimes called \maximal", but that word can easily be misinterpreted to mean \of maximum size". 4 (2) The literature is not consistent on the usage of the term \ranked". Sometimes \ranked" is used for the weaker condition that for every pair x; y 2 P , every chain from x to y has the same length. Under this definition, the implication (1.1) fails (proof left to the reader). (3) For any finite poset P (and some infinite ones) one can define r(x) to be the supremum of the lengths of all chains with top element x | but if P is not a ranked poset, then there will be some pair a; b such that b m a but r(y) > r(x) + 1. For instance, in the bounded poset shown below (known as N5), we have 1^ m y, but r(1)^ = 3 and r(y) = 1. z y x Definition 1.7. Let P be a ranked poset with rank function r. The rank-generating function of P is the formal power series X r(x) FP (q) = q : x2P Thus, for each k, the coefficient of qk is the number of elements at rank k. We can now say that the Boolean algebra is ranked by cardinality. In particular, X jSj n FBn (q) = q = (1 + q) : S⊂[n] The expansion of this polynomial is palindromic, because the coefficients are a row of Pascal's Triangle. That ∗ is, Bn is rank-symmetric. In fact, much more is true. For any poset P , we can define the dual poset P by reversing all the order relations, or equivalently turning the Hasse diagram upside down. It's not hard ∼ ∗ to prove that the Boolean algebra is self-dual, i.e., Bn = Bn, from which it immediately follows that it is rank-symmetric. Example 1.8 (The partition lattice). Let Πn be the poset of all set partitions of [n]. E.g., two elements of Π5 are S = f1; 3; 4g; f2; 5g (abbr.: S = 134j25) T = f1; 3g; f4g; f2; 5g (abbr.: T = 13j4j25) The sets f1; 3; 4g and f2; 5g are called the blocks of S. We can impose a partial order on Πn by putting T ≤ S if every block of T is contained in a block of S; for short, T refines S. 1234 123 123|4 124|3 134|2 234|1 12|3413|24 14|23 12|3 1|23 13|2 12|3|4 13|2|4 23|1|4 14|2|3 24|1|3 34|1|2 1|2|3 1|2|3|4 5 • The covering relations are of the form \merge two blocks into one". • Πn is graded, with 0^ = 1j2j · · · jn and 1^ = 12 ··· n. The rank function is r(S) = n − jSj. • The coefficients of the rank-generating function of Πn are the Stirling numbers of the second kind: S(n; k) = number of partitions of [n] into k blocks. That is, n X n−k Fn(q) = FΠn (q) = S(n; k)q : k=1 2 2 3 For example, F3(q) = 1 + 3q + q and F4(q) = 1 + 6q + 7q + q . Example 1.9 (Young's lattice.). A partition is a sequence λ = (λ1; : : : ; λ`) of weakly decreasing positive integers: i.e., λ1 ≥ · · · ≥ λ` > 0. If n = λ1 + ··· + λ`, we write λ ` n and/or n = jλj. For convenience, set λi = 0 for all i > `. Let Y be the set of all partitions, partially ordered by λ ≥ µ if λi ≥ µi for all i = 1; 2;::: . This is an infinite poset, but it is locally finite, i.e., every interval is finite. Its rank-generating function X X X qjλj = qn λ n≥0 λ`n is given by the justly celebrated formula 1 Y 1 : 1 − qk k=1 There's a nice pictorial way to look at Young's lattice. Instead of thinking about partitions as sequence of numbers, view them as their corresponding Ferrers diagrams: northwest-justified piles of boxes whose ith row contains λi boxes. For example, 5542 is represented by the following Ferrers diagram: Then λ ≥ µ if and only the Ferrers diagram of λ contains that of µ. The bottom part of the Hasse diagram of Y looks like this: 6 Definition 1.10.