**Title:** The dual of L(n)

**Speaker:** Olga Stroilova

**Speaker Info:** MIT

**Brief Description:**

**Special Note**:

**Abstract:**

Fix a prime p, and work in the p-complete setting. The spectrum L(n) can be presented as a summand of the Steinberg factor in the classifying spectrum of B(Z/p)^n. Studying the action of the general linear group GL_n(F_p) on the decomposition of the functional dual of B(Z/p)^n given by the Segal conjecture allows us to show that F(L(n), S) is a wedge of just two indecomposables - a copy of L(n) and an L(n-1).This result was also previously independently obtained by Alan Cathcart in his thesis under the guidance of J. Frank Adams.

These methods can supposedly be pushed further to deduce the duals of the L(n)_{-k}'s - Steinberg pieces in Thom spectra of negative multiples of the reduced regular representation. The expressions for these duals could then be used to evaluate the generalized Tate construction at the sphere spectrum.

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