The authors give a new description of the cofree coalgebra on a vector space V over a field k [see

*M. Sweedler*, Hopf algebras (1969;

Zbl 0194.32901)]. Let A be a graded k-algebra, TV the (graded) tensor algebra on V. A graded linear map f of degree zero from A to TV is called representative if for some graded linear maps

${g}_{1},\xb7\xb7\xb7,{g}_{n}$,

${h}_{1},\xb7\xb7\xb7,{h}_{n}$ of degree zero from A to TV,

$f\left(ab\right)={\sum}_{i=1}^{n}\left({g}_{i}a\right)\left({h}_{i}b\right)$ for all a,b in A. This generalizes the notion of representative function on a group G, e.g.,

$A=the$ group algebra kG,

$V=k$, and restrict to G [see

*G. Hochschild*, Introduction to affine algebraic groups (1971;

Zbl 0221.20055)]. Let

${A}_{v}^{0}$ be the k- vector space of representative maps from A to TV. Using the above notation, the authors show that

${\Delta}f={\sum}_{i=1}^{n}{g}_{i}\otimes {h}_{i}$ is uniquely determined by f, and that

${g}_{i}$,

${h}_{i}$ are in

${A}_{v}^{0}$, so that

$({A}_{v}^{0},{\Delta})$ is a coalgebra with counit which is evaluation at the unit element of A. For

$V=k$, this reduces to the usual notion of

${A}^{0}$ [cf. M. Sweedler]. Now let A be the polynomial algebra k[x] with the usual grading, and

$\pi $ the map from

$k{\left[x\right]}_{v}^{0}$ to V which is evaluation at x. The authors show that

$(k{\left[x\right]}_{v}^{0},\pi )$ is the cofree coalgebra on V. By using symmetric functions, the authors also give an analogous construction of the cofree cocommutative coalgebra on V.