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Published in:
2016 | OriginalPaper | Chapter
9 The Representation Theory of Inverse Monoids
Author : Benjamin Steinberg
Published in: Representation Theory of Finite Monoids
Publisher: Springer International Publishing
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Abstract
In this chapter we develop the representation theory of inverse monoids following the approach of the author [Ste06, Ste08] using groupoid algebras and Möbius inversion. In fact, this work has a precursor in the work of Rukolaĭne [Ruk78, Ruk80], who used alternating sums of idempotents to achieve the same effect as Möbius inversion and used Brandt inverse semigroups instead of groupoids. The author only became aware of the work of Rukolaĭne after [Ste06, Ste08] were published. However, our more explicit approach lets one take advantage of the detailed knowledge of the Möbius function for a number of naturally occurring lattices. We will, for instance, exploit this for our simple character theoretic proof of Solomon’s computation [Sol02] of the tensor powers of the natural module for the symmetric inverse monoid.