Immanants and Positivity

   Brendon Rhodes, (University of Michigan)

We define the Temperly-Lieb immanants using the Temperly-Lieb algebra and prove that these immanants are totally nonnegative, Schur nonnegative, and satisfy a natural generalization of Lindstr¬omÕs Lemma. The Temperly-Lieb immanants are also shown to characterize all totally nonnegative linear combinations of products of two complemenary matrix minors. As applications of this theory, we prove combinatorial tests for determining whether a linear combination of products of two complementary minors is totally nonnegative or Schur nonnegative and show that the dimension of the linear span of complementary minor products of an nxn matrix is the nth Catalan number, C_n. Finally, we give several generalizations of results from linear algebra using these immanants. This is joint work with Mark Skandera at Dartmouth College.

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