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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