Let $G$ be a compact group and let $R(G)$ be the representation ring of $G$. Additively, $R(G)$ is generated by the irreducible representations of $G$. Usually one only deals with those representations which are nonnegative integer combinations of the irreducible representations. However, often one has formulas which apply to an arbitrary element of $R(G)$, and so includes virtual representations which (for this question at least) are the elements of $R(G)$ whose decomposition into irreducible components includes negative coefficients.



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> **Question:** Is there any 'natural' interpretation of virtual representations? In particular, aside from the obvious interpretation as the elements of the formal completion of the semiring of ordinary representations to a ring, is there a natural way to view these objects? Especially helpful would be pictures/ideas others use to assign meaning to virtual representations (if any).
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**Motivation:** Often in decomposing various formulas involving characters, virtual representations arise in one way or another. For example, in Lie groups, the notion of a highest weight representation $\rho\_\omega$ can be extended to arbitrary weights $t$ of $G$ via:


$\rho\_{w(t)} = (-1)^w\rho\_t$


In particular, if $t$ is not a dominant weight of $G$ then there is a unique $w$ in the Weyl Group of $G$ such that $w(t)$ is a dominant weight so that $\rho\_t$ is defined for arbitrary weights $t$; note that the Weyl Dimension Formula agrees with this extension. If the length of $w$ is odd, then $\rho\_t$ will have negative dimension from the dimension formula, hence is virtual.


Another simple example is that when considering the action of the Adams operation $\psi^k$ on $R(G)$, one has that $\psi^k(\rho)$ is in general a virtual representation.


It happens that from time to time I come across other instances of virtual representations appearing in equations I am considering, and I always work with them ignoring whether they have a physical interpretation or not, but at the same time it would be more satisfying if I could understand the equations as manifestations of some deeper structure.


**Edit:** Per Qiaochu's comment, yes, virtual representations can be fit into the framework of super-representations. If it is the case that virtual representations are often viewed as super-representations, then perhaps someone could elaborate on why super-representations are so natural and how one works around the dimension mismatches between virtual representations and super-representations, i.e. $dim(\rho\_1\ominus\rho\_2) = dim(\rho\_1)-dim(\rho\_2)$ but the dimension of the corresponding super-representation is $dim(\rho\_1)+dim(\rho\_2)$.



Is there a 'nice' interpretation of virtual representations?
Your question is really about virtual vector spaces: *what is a virtual vector space?*  



Once you know what a virtual vector space is, then there is only a small step to the answer of your question.


There are a few possible answers:  



1• A virtual vector space of a pair of vector spaces. Equivalently, it's a $\mathbb Z/2$-graded vector space. You should think of the pair $(V,W)$ as being the formal difference $V-W$. This approach has the downside that it makes it unclear what an isomorphism between virtual vector spaces should be.


2• A vector space of dimension $n$ is the same thing as a point (sic!) of the topological space $BU(n)$. A virtual vector space is then a point of the space 
$BU:=\mathrm{colim}\_{n\to \infty} BU(n)$. This approach is not geometric at all, but works very well for talking about virtual vector bundles on a space $X$: these are continuous maps $X\to BU$.


3• Fix an infinite dimensional vector space $U$ and a polarization $U=U\_-\oplus U\_+$ (both $U\_-$ and $U\_+$ are infinite dimensional).
A virtual vector space is a (necessarily infinite dimensional) subspace $V\subset U$ such that $V\cap U\_-$ is of finite codimension inside $V+U\_-$.


4• Fix an infinite dimensional Hilbert space $H$. A virtual vector space is a Fredholm operator $F:H\to H$ (i.e., an operator with finite dimensional kernel and cokernel). This is related to definition 1 by assigning to the Fredholm operator $F$ the pair $(\ker(F),\mathrm{coker}(F))$.


5• A virtual vector space is an object of the bounded derived category of vector spaces: i.e., it's a chain complex.


All these definitions (with the exception of 2, for which it's more involved) can be easily adapted to the context of $G$-representation, you just need to replace "infinite dimensional vector space" with "$G$-rep that contains each irrep infinitely often".


