Virtual characters and the character ring $R(G)$ of a finite group

Let $G$ be a finite group. A virtual character of $G$ is an integral linear combination of irreducible complex characters:

$\vartheta ={\sum }_i{n}_i{\chi }_i,n_i\in \mathbb{Z},$

with only finitely many nonzero coefficients. Since irreducible characters are class functions, every virtual character is a class function on $G$ (Class functions and the complex vector space $\mathrm{c}\mathrm{f}(G)$, An irreducible complex character).

The set of all virtual characters is the character ring $R(G)$. Addition is pointwise addition of class functions, and multiplication is the bilinear extension of tensor-product multiplication on honest characters:

$({\chi }_V+{\chi }_W)(g)={\chi }_V(g)+{\chi }_W(g),({\chi }_V{\chi }_W)(g)={\chi }_{V\otimes W}(g)$

(Characters add on direct sums, multiply on tensor products, and conjugate on duals).

• Multiplication stays in the integral span: a tensor product of two finite-dimensional complex representations is finite-dimensional, and repeated application of Maschke's theorem for finite groups over fields whose characteristic does not divide $\mathrm{\mid }G\mathrm{\mid }$ decomposes it into irreducible summands. Its character is therefore a finite sum of irreducible characters with nonnegative integer multiplicities. Bilinearity then handles virtual characters.

• The ordinary characters form a subsemiring of $R(G)$, while $R(G)$ itself is their Grothendieck group.

• By The irreducible complex characters form an orthonormal basis of $\mathrm{c}\mathrm{f}(G)$, the irreducible characters are linearly independent as class functions. Since virtual characters are defined as their integral span, each virtual character has a unique decomposition in that basis.