## Matrix Entry Functions

**Point of post: **In this post we discuss the concept of how representations give rise naturally to a wide variety of elements of the group algebra. Namely, we discuss the matrix entries of the matrix realizations of an irrep.

*Motivation*

We’ve seen that every finite group gives rise naturally to a set of equivalence classes of irreps . Suppose for a second though that for each equivalence class we picked some representative and fixed a basis for ‘s representation space . Then, we canonically have defined a mapping by where when it’s clear which and we’re discussing we simply write . From this we then have defined elements of the group algebra . Namely, if we denote to be the entry of then the mapping is a mapping and thus an element of the group algebra as stated. In this post we discuss some of the important properties of these *matrix entry functions *as they shall prove absolutely crucial in all of the theory to come.

*Induced Matrix Representation *

Let be a finite group, and (as usual) the set of all equivalence classes if irreps on .* *Then select once and for all a representative for each *.* Then if is the representation space for fix once and for all some ordered orthonormal basis . We then define the map (where we’ve denoted the degree of by for notational convenience) by . We call this the matrix realization of . We first claim that each is itself an irrep with representation space *. *Indeed,

**Theorem: ***Let be defined as above. Then is itself and irrep on .*

**Proof: **It’s evident that . Thus, to verify that is a representation of it suffices to show that is a homomorphism. But, using the fact that the map is an associative unital algebra isomorphism we know then that

from where the conclusion follows. Now, to see that is irreducible let be the canonical identification (the isomorphism associating with each element of the column vector of the coefficients of that element with respect to the ordered basis). Recall then that and let be -invariant. Note then that for any

and since was arbitrary it follows that is -invariant and so is equal to either or which is equivalent to saying that is equal to either or . Thus, since was arbitrary it follows that is irreducible.

**References:**

1.Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

2. Serre, Jean Pierre. *Linear Representations of Finite Groups*. New York: Springer-Verlag, 1977. Print.

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