Increasing Subsequences and the Classical Groups

We show that the moments of the trace of a random unitary matrix have combinatorial interpretations in terms of longest increasing subsequences of permutations. To be precise, we show that the 2n-th moment of the trace of a random k-dimensional unitary matrix is equal to the number of permutations of length n with no increasing subsequence of length greater than k. We then generalize this to other expectations over the unitary group, as well as expectations over the orthogonal and symplectic groups. In each case, the expectations count objects with restricted “increasing subsequence” length. Introduction Much work has been done in the combinatorial literature on the “increasing subsequence problem”, that of studying the distribution of the length of the longest increasing subsequence of a random permutation. The problem was first considered by Hammersley ([5]); good summaries can be found in [1] and [10], which gives an alternate proof of Theorem 1.1. This problem is also closely connected to the representation theory of Sn, particularly the theory of Young tableaux. The representation theory aspects are covered in [13]; section 5.1.4 in [8] gives a good treatment of the more elementary Young tableaux results. The results reported here arose from the observation that a certain partial sum of characters of the symmetric group that occurs naturally in the increasing subsequence problem also appears when calculating certain expectations over the unitary group. In particular, it turns out that the distribution of the length of the longest increasing subsequence can be expressed exactly in terms of the moments of the trace of a random (uniformly distributed) unitary matrix. This correspondence generalizes both to other moments for the unitary group, and to the moments of the trace of a random orthogonal or symplectic matrix. In each case, the moments count objects (colored permutations, signed permutations, or fixed-point-free involutions) with restricted increasing subsequence length. Section 1 states and proves the connection between the classical increasing subsequence problem and the unitary group. Section 2 extends this to other increasing subsequence problems connected to the unitary group, including an increasing subsequence problem for signed permutations (the hyperoctahedral group). Section 1991 Mathematics Subject Classification. Primary 05E15, Secondary 05A15 05A05.