Exercise set 2.4

Throughout, K denotes a field. We denote:

• by I(n,m) the set of maps from [n] to [m],
• by OP(n,m) the set of partitions of [n] into m ordered parts.

These sets come with natural actions of 𝔖_n on them.

(1)

(Difficulty level 1) Let X be a finite set on which a group G acts. Show that KX (the free K-vector space generated by X) and K[X] (the space of K-valued functions on X) are isomorphic as G-representations via xδ_x.

(2)

(Difficulty level 1) Show the following: for elements p and q of I(n,m) and A, B matrices of size m×m (with entries over some field or commutative ring), (AB)_p,q = ∑ _t∈I(n,m)A_p,t⋅B_t,q.

(3)

(Difficulty level 1) Describe a “natural” bijection between I(n,m) and OP(n,m) that preserves type and is 𝔖_n-equivariant.

(4)

(Difficulty level 1) Let λ be a weak composition of n into m parts. Let λ¹ be the partition of n into at most m parts obtained by putting the constituents of λ in weakly decreasing order. Let X_λ be the set of all ordered partitions of n into m parts with type λ. Observe that X_λ and X_λ¹ are isomorphic as 𝔖_n-sets.

(5)

(Difficulty level 3) Work out the multiplicative structure constants of SK(m,n) for some small values of m and n.