Expected length of a product of random reflections

We present a simple formula for the expected number of inversions in a permutation of size $n$ obtained by applying $t$ random (not necessarily adjacent) transpositions to the identity permutation. More general, for any finite irreducible Coxeter group belonging to one of the infinite families (type A, B, D, and I), an exact expression is obtained for the expected length of a product of $t$ random reflections.


Introduction
In 2000, Eriksson et. al. [6] studied the expected number of inversions after t random adjacent transpositions applied to the identity permutation. Their main motivation came from the area of molecular evolution where a genome (modelled as a permutation) undergoes random mutations (modelled as adjacent transpositions). Since then several people have analysed this and similar problems, motivated both by biology and pure mathematics.
In 2005, Eriksen [3] improved the results from [6] and obtained an exact formula for the expected number of inversions. More recently, in 2010, Bousquet-Mélou [2] gave a completely different exact expression for the same thing! In 2002, Troili [7] generalized the question to other groups than the symmetric group, and she gave corresponding exact formulas for the dihedral groups I 2 (m).
In 2004, Eriksen and Hultman [4] modelled the genome rearrangement problem closer to reality by considering any random transpositions (and not just adjacent ones) as mutations. Instead of the number of inversions they computed the expected absolute length of the resulting permutation, i.e. the minimal number of transpositions whose product is the permutation. In a sequel [5] they generalized their results to a class of complex reflection groups.
For an exposé of the research on more general aspects of Markov chains generated by random (adjacent) transpositions, such as mixing times, we refer to Bousquet-Mélou [2].
In the present paper, we will generate permutations in the same way as Eriksen and Hultman, i.e. by random transpositions, but we will meausure them by the number of inversions as in the original papers. We will also generalize the framework to all finite Coxeter groups and present exact expectancy formulas for the groups A n , B n , D n and I 2 (m). In Coxeter group lingo, we compute the expected length of a product of t random reflections.
The paper is organized as follows. After a section with some basic definitions, we review the previously known results mentioned above so that the reader gets a feeling for what kind of expressions may appear for the expected (absolute) length and also what type of combinatorics is going on behind the formulas. Next, in section 4 we loosen up by computing the expected length and absolute length of a product of random reflections or simple reflections in the dihedral groups I 2 (m). After this relatively simple task, in section 5 we are ready to state and prove our main results: formulas for the expected length of a product of random reflections in the Weyl groups of type A, B, and D. Finally, in section 6 we summarize the current state of affairs and reveal some of our ideas for future research.

Definitions and notation
We will follow the notation from the book by Björner and Brenti [1] where also the definitions of the basic concepts may be found. For the reader's convenience we have included the most relevant definitions below.
A Coxeter group is a group W with a presentation of the form where m is a function from S to {1, 2, . . . , ∞} such that m(s, s ′ ) = m(s ′ , s) for all s, s ′ ∈ S and m(s, s ′ ) = 1 if and only if s = s ′ . The elements of S (which we identify with their image in W through the presentation) are called generators or simple reflections and the pair (W, S) is called a Coxeter system. The group elements that are conjugate to some simple reflection are called reflections and the set of reflections is usually denoted by T . By definition, any group element w ∈ W is a (finite) product of simple reflections, and the minimal ℓ such that w is a product of ℓ simple reflections is called the length of w and is denoted by ℓ(w). Similarly, the minimal ℓ ′ such that w is a product of ℓ ′ reflections is called the absolute length of w and is denoted by ℓ ′ (w). Now, let us make a very general definition. to be the expected "length" of a product of t random elements, chosen independently and uniformly from R.
As a special case, for a Coxeter system (W, S) with S finite, E W S,ℓ (t) denotes the expected length of a product of t simple reflections. Similarly, if W is finite, E W T,ℓ ′ (t) denotes the expected absolute length of a product of t reflections.
As mentioned in the introduction, exact formulas for E W S,ℓ (t) have been found for the symmetric groups (W, S) = A n by Eriksen [3] and for the dihedral groups (W, S) = I 2 (m) by Troili [7]. Exact formulas for E W T,ℓ ′ (t) have been found by Eriksen and Hultman, first for the symmetric groups [4] and later for the complex reflection groups G(r, 1, n) where the hyperoctahedral groups B n are included.
In the present paper we will compute E W T,ℓ for the three infinite families of finite irreducible Coxeter systems A n , B n , D n , and I 2 (m).

Known results
To get a flavour of the kind of expressions that may appear, we will begin by presenting the already known results and discussing briefly what methods were used to obtain them, before we reveal our own results in the next section. .
The proof of the first formula, due to Eriksen, is built on a discrete heat flow process in a two-dimensional lattice. His expression has a combinatorial interpretation in terms of lattice walks. The second formula, due to Bousquet-Mélou, was obtained by attacking the same heat flow process by generating functions and the kernel method. It has the advantage that the dependence on t is very explicit which makes it possible to analyse what happens in the limit when n and t tend to infinity at the same time in various ways.
Theorem 3.2 (Troili, 2002). Let m ∈ {2, 3, . . . , ∞} and let t ≥ 0. Then the expected length of a product of t random simple reflections in the dihedral group I 2 (m) is Here, binomial coefficients a b with b negative (or b = −∞) vanish by definition. Troili's proof is built on the observation that where π (r) is a product of r random reflections in I 2 (m). Theorem 3.3 (Eriksen, Hultman, 2005). Let r, n be positive integers not both equal to one. Then the expected absolute length of a product of t random reflections in the complex reflection group G(r, 1, n) is where The proof of Eriksen and Hultman's formula involves group representations and characters. To see why this is natural, think of the symmetric group A n−1 where the absolute length of a permutation is n minus the number of cycles in the cycle representation of the permutation. Applying a random transposition either splits a cycle or merges two cycles, and it is easy to see that during this process we only have to keep track of the cycle type of the permutation, i.e. its conjugacy class.

The dihedral groups
As we have seen, computing E I 2 (m) S,ℓ (t) for the dihedral groups lead to a rather cumbersome expression in Theorem 3.2. In this section, we will see that much nicer formulas exist for E T,ℓ (t) of a product of t random reflections in the dihedral group I 2 (m) is m/2 (for all positive t) if m is even and m 2 − (−1) t 2m if m is odd. Proof. Note that in I 2 (m) the set T of reflections is precisely the set of elements with odd length. Thus, multiplying any element in T by a uniformly chosen random reflection yields a uniformly chosen random element in I 2 (m) \ T , and, similarly, multiplying any element in I 2 (m) \ T by a uniformly chosen random reflection yields a uniformly chosen random element in T . A simple calculation reveals that the average length of an element in T is m/2 if m is even and m 2 + 1 2m if m is odd, and that the average length of an element in if m is odd. Proof. Clearly, in a dihedral group, the absolute length of an element w is 1 if ℓ(w) is odd and 2 if ℓ(w) is even, unless w = id. The probability that a product of an even number t of simple reflections is the identity element is precisely the sum above divided by 2 t .

The Weyl groups of type A, B and D
In this section we will obtain exact expressions for E W T,ℓ (t) when W is a Weyl group of type A, B, and D. The formulas and their proofs turn out to be quite similar for these three types, but the symmetric groups happen to be a bit less technical to deal with, so we start with those.
Theorem 5.1. For n ≥ 2 and t ≥ 0, let π (t) ∈ A n−1 be the random permutation obtained by applying t random transpositions to the identity permutation. Then the following holds.
(a) The expected number of inversions in π (t) is Proof. Part (a) of the theorem follows from part (b) by the simple observation that 1≤i<j≤n (j − i) = n(n 2 − 1)/6. It remains to prove part (b). For every 1 ≤ i, j ≤ n and t ≥ 0, let Clearly, u i,i = 0 and for i = j, j,i for 1 ≤ i, j ≤ n. The above recurrence for u translates to the following recurrence for v.
Using that v (t) i,i = 0 for any i, we can write the recurrence in a more elegant way: Next, we will express the above recurrence in terms of linear operators. Let A n denote the vector space over R of all real antisymmetric n × n matrices, and define the linear operator Q on A n as follows. For any v ∈ A n and for any 1 ≤ i, j ≤ n, let each entry in Qv be its row sum plus its and column sum in v, or formally, Now, the recurrence relation (2) can be written where I is the identity operator on A n and x = n(n−5)

2
. The starting point v (0) is the n × n matrix with zeroes on the main diagonal, ones above it and minus ones below it. From the simple but crucial observation that Q 2 = nQ we deduce that A striking feature of the second part of Theorem 5.1 is that, for a fixed t, the probility that i < j is an inversion depends only on the difference j − i. This fact is not evident from the definition of the random process, but nevertheless it can be proved without invoking Theorem 5.1.
By symmetry of the random process, we have Prob(π i > n−k) = Prob(π j > n−k) and hence Prob(π i > π j ) = Prob(π +k i > π +k j ). Equation (3) now follows from the observation that Prob(π i+k > π j+k ) = Prob(π +k i > π +k j ). 5.2. A definition and a lemma. The situation becomes a little more delicate for Weyl groups of type B and D, but in a way that is very similar for these types. To be economical we gather the bit of reasoning that is common for type B and D in a definition and a lemma.
together with the initial condition v Proof. Let D n denote the real vector space of all functions v : (We write v i,j instead of the more common v(i, j) since we may view D n as the space of 2n × 2n matrices that are antisymmetric about the main diagonal and symmetric about the antidiagonal, if we disregard the entries on the diagonals.) Let Q be the linear operator on D n defined by for any v ∈ D n and (i, j) ∈ I n , We have (now using the symmetry and antisymmetry properties) Thus, we have showed that Q 2 = (2n − 2)Q which means that where I is the identity operator on D n . The lemma now follows from the observation that

The hyperoctahedral groups.
Recall that B n is isomorphic to the subgroup of permutations π of the set [−n, n] \ {0} such that π −i = −π i for all 1 ≤ i ≤ n (see e.g. [1, Ch. 8]). In this representation, the set of reflections is in cycle notation. The length of π equals its number of B-inversions, which are pairs (i, j) with i, j ∈ [−n, n] \ {0} and j ≥ |i| such that π i > π j .
Theorem 5.5. For n ≥ 1 and t ≥ 0, let π (t) be a product of t random reflections in B n . Then the following holds.
(a) The expected length of π (t) is For 1 ≤ i ≤ n, the probability that π Proof. Part (a) of the theorem follows from part (b) by the simple observation that j>|i| (j − i) = 2n(n 2 − 1)/3. It remains to prove part (b). For every i, j ∈ [−n, n] \ {0} and t ≥ 0, let i,j . The following recurrence relation holds.
We also have the symmetry property u j,i for i, j ∈ [−n, n] \ {0}. Clearly, v inherits the symmetry property from u and has also the antisymmetry property v (t) i,j . The above recurrence for u is valid also for v and using the symmetry and antisymmetry properties it can be written Clearly, this implies that v After dividing this by n 2t we obtain . This proves part (b) of the theorem.

The groups of type D.
Recall that D n is isomorphic to the subgroup of permutations π of the set [−n, n] \ {0} such that π −i = −π i for all 1 ≤ i ≤ n and there is an even number of i ∈ {1, 2, . . . , n} such that π i < 0 (again see e.g. [1,Ch. 8]). In this representation, the set of reflections is in cycle notation. The length of π equals its number of D-inversions, which are pairs (i, j) with i, j ∈ [−n, n] \ {0} and j > |i| such that π i > π j .
Theorem 5.6. For n ≥ 1 and t ≥ 0, let π (t) be a product of t random reflections in D n . Then the following holds.
(a) The expected length of π (t) is Proof. Part (a) of the theorem follows from part (b) by the simple observation that j>|i| (j − i) = 2n(n 2 − 1)/3. It remains to prove part (b). For every t ≥ 0 and (i, j) ∈ I n , let The following recurrence relation holds: We also have the symmetry property u j,i for (i, j) ∈ I. Clearly, v (t) inherits the symmetry property from u (t) and has also the antisymmetry property v (t) i,j . The above recurrence for u is valid also for v and using the symmetry and antisymmetry properties it can be written By Lemma 5.4 we obtain, for j > |i|, Dividing by (n 2 − n) t yields This proves part (b) of the theorem.

Conclusion and ideas for future research
We can sum up the current state of knowledge by the following table, listing the groups W for which we have exact expressions for E W R,ϕ (t).
R = simple reflections R = all reflections ϕ = length A n , I 2 (m), I 2 (∞) A n , B n , I 2 (m), G(r, 1, n) ϕ = absolute length I 2 (m), I 2 (∞) A n , B n , D n , I 2 (m) Perhaps the two most striking things with this table are that B n is missing from the upper left square and that not even A n appears in the lower left square! As Troili has pointed out [7], the problem of computing E Bn S,ℓ (t) can be reduced to a heat flow process similar to that of the symmetric group, but it is not clear how to analyze it further.
Computing E An S,ℓ ′ (t) is equivalent to computing the expected number of cycles (in the cycle representation of the permutation) after applying t random adjacent transpositions to the identity permutation. For now, we have no idea how to attack this natural problem. Note that it is not enough to keep track of the conjugacy class of the permutation as Eriksen and Hultman did in the proof of Theorem 3.3.
It is also worth noting that E W S,ℓ (t) and E W S,ℓ ′ (t) is well-defined for any finitely generated Coxeter system. (Indeed, Troili made an effort to compute EÃ n S,ℓ (t) in [7], but unfortunately with an erraneous heat flow process.) In addition to the length and the absolute length, there is at least one more "length" function that comes naturally with any Coxeter system (W, S), namely the descent number defined by d(w) = #{s ∈ S : ℓ(ws) < ℓ(w)}.
(In fact, the ordinary length function is also a kind of descent number by the equality ℓ(w) = #{t ∈ T : ℓ(wt) < ℓ(w)}, see e.g. [1,Corollary 1.4.5].) What is E W S,d (t) and E W T,d (t) for a (in the second case, finite) Coxeter system (W, S)?

Acknowledgement
This work was performed at KTH in Stockholm and was supported by a grant from the Swedish Research Council (621-2009-6090).