Centralizer ring
Definition
Definition with symbols
Given a group , a subgroup , and a ring , the centralizer ring of with respect to the subgroup and over the ring is defined in any of the following equivalent ways:
- It is the endomorphism ring of the -module where the module action is defined by coordinate-wise left multiplication by
- It is the endomorphism ring of the -module where acts on the coset space by left multiplication.
- It is the endomorphism ring of the -module over the double coset space of in .
![{\displaystyle R[G]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cfc4570ea0faa6453735827f1e877904ecf78fe)
![{\displaystyle R[G/B\times G/B]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e9c4ec32a11ef879921262192f9eef12e1f6810)
![{\displaystyle R[B]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ba85352cf7fd9f17e6c6e1f41f47975387ea6c8)
![{\displaystyle R[G/B]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5a8dc40ee33af92071cdcb6f53d4dbf09446642)
In the particular case where is an algebraic group and is a Borel subgroup, the corresponding centralizer ring is called the Hecke algebra.