Reduced words and a formula of Macdonald

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1 Reduced words and a formula of Macdonald Sara Billey University of Washington Based on joint work with: Alexander Holroyd and Benjamin Young preprint arxiv: Graduate Student Combinatorics Conference 07 April 8, 07

2 Outline Permutations and Reduced Words Macdonald s Reduced Word Formula Generalizations of Macdonald s Formula Open Problems

3 Permutations Permutations are fundamental objects in mathematics, computer science, game theory, economics, physics, chemistry and biology. Notation. S n is the symmetric group of permutations on n letters. w S n is a bijection from [n] := {,,..., n} to itself denoted in one-line notation as w = [w(), w(),..., w(n)]. s i = (i i + ) = adjacent transposition for i < n. Example. w = [,,,, 5] S 5 and s = [,,, 5, ] S 5. ws = [,,, 5, ] and s w = [, 5,,, ].

4 Permutations Presentation of the Symmetric Group. Fact. S n is generated by s, s,..., s n with relations s i s i = (s i s j ) = if i j > (s i s i+ ) = For each w S n, there is some expression w = s a s a s ap. If p is minimal, then l(w) = length of w = p, s a s a s ap is a reduced expression for w, a a... a p is a reduced word for w.

5 Reduced Words and Reduced Wiring Diagrams Example. and are reduced words for [,, ]. Example. 565 is a reduced word for [,, 6, 5, 7,, ] S

6 Reduced Words Key Notation. R(w) is the set of all reduced words for w. Example. R([,, ]) = {, }. Example. R([,,, ]) has 6 elements: Example. R([5,,,, ]) has 768 elements.

7 Counting Reduced Words Question. How many reduced words are there for w?

8 Counting Reduced Words Question. How many reduced words are there for w? Theorem.(Stanley, 98) For w n 0 := [n, n,...,, ] S n, R(w n 0 ) = ( n )! n n 5 n (n ). Observation: The right side is equal to the number of standard Young tableaux of staircase shape (n, n,..., ).

9 Counting Standard Young Tableaux Defn. A partition of a number n is a weakly decreasing sequence of positive integers such that n = λ i = λ. λ = (λ λ λ k > 0) Partitions can be visualized by their Ferrers diagram (6, 5, ) Def. A standard Young tableau T of shape λ is a bijective filling of the boxes by,,..., n with rows and columns increasing. Example. T = The standard Young tableaux (SYT) index the bases of S n -irreps.

10 Counting Standard Young Tableaux Hook Length Formula.(Frame-Robinson-Thrall, 95) If λ is a partition of n, then n! #SYT (λ) = c λ h c where h c is the hook length of the cell c, i.e. the number of cells directly to the right of c or below c, including c. Example. Hook lengths of λ = (5,, ): 7 5 9! So, #SYT (5,, ) = 7 5 = 6. Remark. Notable other proofs by Greene-Nijenhuis-Wilf 79 (probabalistic), Krattenthaler 95 (bijective), Novelli-Pak-Stoyanovskii 97 (bijective), Bandlow 08.

11 Counting Reduced Words Theorem.(Edelman-Greene, 987) For all w S n, R(w) = a λ,w #SYT (λ) for some nonnegative integer coefficients a λ,w with λ l(w) in a given interval in dominance order. Proof via an insertion algorithm like the RSK: a = a a... a p (P(a), Q(a)). P(a) is strictly increasing in rows and columns has reading word equal to a reduced word for w. Q(a) can be any standard tableau of the same shape as P(a). Corollary. Every reduced word for w 0 inserts to the same P tableau of staircase shape δ, so R(w 0 ) = #SYT (δ).

12 Random Reduced Word The formula R(w) = a λ,w #SYT (λ) gives rise to an easy way to choose a random reduced word for w using the Hook Walk Algorithm (Greene-Nijenhuis-Wilf) for random STY of shape λ. Algorithm. Input: w S n, Output: a a... a p R(w) chosen uniformly at random.. Choose a P-tableau for w in proportion to #SYT (sh(p)).. Set λ = sh(p).. Loop for k from n down to. Choose one of the k empty cells c in λ with equal probability. Apply hook walk from c.. Hook walk: If c is in an outer corner of λ, place k in that cell. Otherwise, choose a new cell in the hook of c uniformly. Repeat step until c is an outer corner.

13 Random Reduced Word Def. For a = a a... a p R(w), let B(a) be the random variable counting the number of braids in a, i.e. consecutive letters i, i +, i or i +, i, i +. Examples. B() = and B() = Question. What is the expected value of B on R(w)?

14 Random Reduced Word Def. For a = a a... a p R(w), let B(a) be the random variable counting the number of braids in a, i.e. consecutive letters i, i +, i or i +, i, i +. Examples. B() = and B() = Question. What is the expected value of B on R(w)? Thm.(Reiner, 005) For all n, the expected value of B on R(w 0 ) is exactly.

15 Random Reduced Word Angel-Holroyd-Romik-Virag: Random Sorting Networks (007) Conjecture. Assume a a... a p R(w 0 ) is chosen uniformly at random. The distribution of s in the permutation matrix for w = s a s a s ap/ converges as n approaches infinity to the projected surface measure of the -sphere.

16 Random Reduced Word Alexander Holroyd s picture of a uniformly random 000-element sorting network (selected trajectories shown):

17 Macdonald s Formula Thm.(Macdonald, 99) For w 0 S n, a R(w 0 ) a a a ( n ) =

18 Macdonald s Formula Thm.(Macdonald, 99) For w 0 S n, a R(w 0 ) a a a ( n ) = ( ) n!

19 Macdonald s Formula Thm.(Macdonald, 99) For w 0 S n, a R(w 0 ) a a a ( n ) = ( ) n! Question.(Holroyd) Is there an efficient algorithm to choose a reduced word randomly with P(a a... a ( n ) proportional to ) a a a ( n )?

20 Consequences of Macdonald s Formula Thm.(Young, 0) There exists a Markov growth process using Little s bumping algorithm adding one crossing in a wiring diagram at a time to obtain a random reduced word for w 0 S n in ( n) steps. Image credit: Kristin Potter.

21 Consequences of Macdonald s Formula The wiring diagram for a random reduced word for w 0 S 600 chosen with Young s growth process.

22 Macdonald s Formula The permutation matrix for the product of the first half of a random reduced word for w 0 S 600 chosen with Young s growth process.

23 Macdonald s Formula Thm.(Macdonald, 99) For any w S n with l(w) = p, a R(w) a a a p = p!s w (,,,...) where S w (,,,...) is the number of monomials in the corresponding Schubert polynomial. Question.(Young, Fomin, Kirillov,Stanley, Macdonald, ca 990) Is there a bijective proof of this formula?

24 Macdonald s Formula Thm.(Macdonald, 99) For any w S n with l(w) = p, a R(w) a a a p = p!s w (,,,...) where S w (,,,...) is the number of monomials in the corresponding Schubert polynomial. Question.(Young, Fomin, Kirillov,Stanley, Macdonald, ca 990) Is there a bijective proof of this formula? Answer. Yes! Based on joint work with Holroyd and Young, and builds on Young s growth process.

25 Schubert polynomials History. Schubert polynomials were originally defined by Lascoux-Schützenberger early 980 s. Via work of Billey-Jockusch-Stanley, Fomin-Stanley, Fomin-Kirillov, Billey-Bergeron in the early 990 s we know the following equivalent definition. Def. For w S n, S w (x, x,... x n ) = D RP(w) x D where RP(w) are the reduced pipe dreams for w, aka rc-graphs. Example. A reduced pipe dream D for w = [, 6,,, 5, ] where x D = x x x x

26 Bijective Proof of Macdonald s Formula To show: a R(w) a a a p = p! #RP(w)

27 Bijective Proof of Macdonald s Formula To show: a R(w) a a a p = p! #RP(w) Def. b b... b p is a bounded word for a a... a p provided b i a i for each i. Def. The pair (a, b) = ((a a... a p ), (b b... b p )) is a bounded pair for w provided a R(w) and b is a bounded word for a. Def. A word c = c c... c p is a sub-staircase word provided c i i for each i.

28 Bijective Proof of Macdonald s Formula To show: a R(w) a a a p = p! #RP(w) Want: A bijection BP(w) cd(w) where BP(w) := bounded pairs for w, cd(w) := cd-pairs for w of the form (c, D) where D is a reduced pipe dream for w and c is a sub-staircase word of the same length as w.

29 Bijective Proof of Macdonald s Formula a: b: c: D: M M M M M

30 Transition Equations Thm.(Lascoux-Schützenberger,98) For all w id, let (r < s) be the largest pair of positions inverted in w in lexicographic order. Then, S w = x r S wtrs + S w where the sum is over all w such that l(w) = l(w ) and w = wt rs t ir with 0 < i < r. Call this set T (w).

31 Transition Equations Thm.(Lascoux-Schützenberger,98) For all w id, let (r < s) be the largest pair of positions inverted in w in lexicographic order. Then, S w = x r S wtrs + S w where the sum is over all w such that l(w) = l(w ) and w = wt rs t ir with 0 < i < r. Call this set T (w). Example. If w = [756], then r = 5, s = 7 S w = x 5 S [756] + S [756] + S [756] So, T (w) = {[756], [756]}.

32 Little s Bijection Theorem.(David Little, 00) There exists a bijection from R(w) to w T (w)r(w ) which preserves the ascent set provided T (w) is nonempty. Theorem. (Hamaker-Young, 0) Little s bijection also preserves the Coxeter-Knuth classes and the Q-tableaux under the Edelman-Greene correspondence. Furthermore, every reduced word for any permutation with the same Q tableau is connected via Little bumps.

33 Little Bumps Example. The Little bump applied to a = 565 in col.

34 Push and Delete operators Let a = a... a k be a word. Define the decrement-push, increment-push, and deletion of a at column t, respectively, to be P t a = (a,..., a t, a t, a t+,..., a k ); P + t a = (a,..., a t, a t +, a t+,..., a k ); D t a = (a,..., a t, a t+,..., a k );

35 Bounded Bumping Algorithm Input: (a, b, t 0, d), where a is a word that is nearly reduced at t 0, and b is a bounded word for a, and d {, +}. Output: Bump d t 0 (a, b) = (a, b, i, j, outcome).. Initialize a a, b b, t t 0.. Push in direction d at column t, i.e. set a P d t a and b P d t b.. If b t = 0, return (D t a, D t b, a t, t, deleted) and stop.. If a is reduced, return (a, b, a t, t, bumped) and stop. 5. Set t Defect t (a ) and return to step.

36 Generalizing the Transition Equation. We use the bounded bumping algorithm applied to the (r, s) crossing in a reduced pipe dream for w to bijectively prove S w = x r S wtrs + S w.. We use the bounded bumping algorithm applied to the (r, s) crossing to give a bijection BP(w) BP(wt rs ) [, p] w T (w) BP(w ).

37 Bijective Proof of Macdonald s Formula a R(w) a a a p = p!s w (,,,...) a: b: c: D: M M M M M

38 q-analog of Macdonald s Formula Def. A q-analog of any integer sequence f, f,... is a family of polynomials f (q), f (q),... such that f i () = f i. Examples. The standard q-analog of a positive integer k is [k] = [k] q := + q + q + + q k. The standard q-analog of the factorial k! is defined to be [k] q! := [k][k ] []. Macdonald conjectured a q-analog of his formula using [k], [k] q!.

39 q-analog of Macdonald s Formula Theorem.(Fomin and Stanley, 99) Given a permutation w S n with l(w) = p, a R(w) [a ] [a ] [a p ] q comaj(a) = [p] q! S w (, q, q,...) where comaj(a) = i. a i <a i+ Remarks. Our bijection respects the q-weight on each side so we get a bijective proof for this identity too. The key lemma is a generalization of Carlitz s proof that l(w) and comaj(w) are equidistributed on S n and another generalization of the Transition Equation.

40 Another generalization of Macdonald s formula Fomin-Kirillov, 997. We have the following identity of polynomials in x for the permutation w 0 S n : a R(w 0 ) (x + a ) (x + a ( n ) ) = ( ) n! i<j n x + i + j. i + j Remarks. Our bijective proof of Macdonald s formula plus a bijection due to Lenart, Serrano-Stump give a new proof of this identity answering a question posed by Fomin-Kirillov. The right hand side is based on Proctor s formula for reverse plane partitions and Wach s characterization of Schubert polynomials for vexillary permutations.

41 Open Problems Open. Is there a common generalization for the Transition Equation for Schubert polynomials, bounded pairs, and its q-analog? Open. Is there a nice formula for rpp λ (x) or [rpp λ (x)] q for an arbitrary partition λ as in the case of staircase shapes as noted in the Fomin-Kirillov Theorem? Open. What is the analog of Macdonald s formula for Grothendieck polynomials and what is the corresponding bijection?