View source for Ring of representations

''representation ring''

A [[Commutative ring|commutative ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r0824201.png" /> defined as follows. The additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r0824202.png" /> is generated by the equivalence classes of linear representations of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r0824203.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r0824204.png" />, and the defining relations have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r0824205.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r0824206.png" /> is the equivalence class of a representation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r0824207.png" /> is the equivalence class of a subrepresentation of it and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r0824208.png" /> is the equivalence class of the corresponding quotient representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r0824209.png" />; the multiplication of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r08242010.png" /> assigns to two equivalence classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r08242011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r08242012.png" /> the equivalence class of their tensor product. This ring of representations is sometimes called the Grothendieck ring of the group. For locally compact groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r08242013.png" /> the representation ring is usually meant to be the commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r08242014.png" /> defined by the operations of direct sum and tensor product in the set of equivalence classes of continuous unitary representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r08242015.png" />. The structures of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r08242016.png" /> are very useful if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r08242017.png" /> is compact. Then it leads to the duality theory in terms of block-algebras. In the more general case of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r08242018.png" /> of type I, the study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082420/r08242019.png" /> may be reduced to the study of tensor products of irreducible unitary representations.



====Comments====
One must impose some finiteness condition on the representations under consideration, as otherwise the representation ring will be the zero ring.

A second representation ring is obtained by considering equivalence classes of suitable representations modulo split short exact sequences (instead of short exact sequences). Unless the class of representations involved consists totally of completely-reducible representations (as in the case of compact groups) the two rings of representations can be quite different.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.W. Curtis,   I. Reiner,   "Methods of representation theory" , '''II''' , Wiley (Interscience)  (1987)</TD></TR></table>