# Mathematics Colloquia and Seminars

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### Increasing subsequences and invariant theory

**Algebra & Discrete Mathematics**

Speaker: | Dr. Eric Rains, CalTech |

Location: | 693 Kerr |

Start time: | Fri, Oct 18 2002, 12:00PM |

One of the nicer connections between increasing subsequences of permutations and random matrices is the following fact: the expectation of |Tr(U)|^{2n}, where U is uniformly distributed from U(k), is exactly given by the number of permutations of length n with no increasing subsequence of length greater than k. I'll discuss a refinement of this result, based on the observation that such an expectation can be interpreted as the dimension of a certain space of invariants: the refinement then gives an explicit basis of this space indexed by permutations without long increasing subsequences. This generalizes in a number of ways, both by taking a more complicated integrand (which replaces "permutation" with "multiset" with a suitable notion of increasing subsequence), and by changing the group (which imposes corresponding symmetry conditions on the permutation/multiset). One consequence is an analogue of the classical straightening algorithm for the orthogonal and symplectic groups.

For details please contact Craig Tracy or Jesus De Loera.