Matroid intersection, pointer chasing, and Young's seminormal representation of Sn

We consider the number of queries needed to solve the matroid intersection problem, a question raised by Welsh (1976). Given two matroids of rank r on n elements, it is known that O(nr) independence queries suffice. However, no non-trivial lower bounds are known for this problem. We make the first progress on this question. We describe a family of instances of rank r = n/2 based on a pointer chasing problem, and prove that (log2 3)n−o(n) queries are necessary to solve these instances. This gives a constant factor improvement over the trivial lower bound of n for matroids of this rank. Our proof uses methods from communication complexity and group representation theory. We analyze the communication matrix by viewing it as an operator in the group algebra of the symmetric group and explicitly computing its spectrum.