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Composition of species is closely related to the composition of symmetric collections of vector spaces ("S-modules"), which is a remarkable example of a monoidal category everyone who had ever encount …

Surely there are many: these are all polynomials in one variable, so every two of them are algebraically dependent because of the transcendence degree argument :-) However, I am sure that this is not …

Hermite's reciprocity: as representations of $GL_2$, we have $$ S^k(S^l\mathbb{C}^2)\simeq S^l(S^k\mathbb{C}^2). $$

This is a bit of cheating (since requires to know a bit of homological algebra) but is too nice to not be mentioned: Let's notice that the generating function for dimensions of graded components fo …

Gjergji Zaimi already said it all, but I want to point out a tiny bit longer but equally cute way to derive the same formula. Every monic polynomial over $\mathbb{F}\_q$ decomposes into a product of i …

A natural question covering both this and this question would be Let $n>2$. Describe Young diagrams $\lambda$ with at most $n$ nonempty rows (or equivalently non-increasing sequences $\lambda=(\lamb …

I recall seeing in various sources the terminology "cover preserving embedding" and "cover preserving subposet". Googling it now (https://www.google.com/search?q=poset+%22cover+preserving%22) brings s …

This is very standard. For a compact complex variety admitting a cell decomposition, the (co)homology is the free Abelian group generated by the cells (over $\mathbb{C}$ there is no room for the diffe …

To offer a slightly more geometric viewpoint on the same, the space $\bigoplus_q \mathop{\mathrm{Hom}}_K(\Lambda^q(\mathfrak{p}),\mathbb{C})$, which is the direct sum of all spaces you are considering …

No, it is not multiplicity-free. Already for $d=6$, this representation contains the Schur functor $S^{4,4,4,2,2,2}$ twice. This can be easily checked in Magma (even the online calculator) issuing the …

The argument goes as follows. Let us consider the events $A_i=\{ i(i+1) \text{ occurs in a permutation} \}$ and $B_i=\{ (i+1)i \text{ occurs in a permutation} \}$. Some pairs of events like that canno …

Depending on whether you want it to agree with the symmetric structure or only with monoidal structure, this would be usually referred to, respectively, as twisted commutative algebras or twisted asso …

The answer for $L_1=\sum_ix_i^2\partial_i$ can be derived in a rather straightforward way (I changed your convention a little bit to match the usual formulas for Virasoro algebra). Namely, use the det …

Q1: This algebra is just the Manin black product of $A$ and $A^!$ (in other words, the Koszul dual of the Segre product of $A$ and $A^!$), and hence it is Koszul. (As requested, the Segre product of t …

The following statement seems to not have clear combinatorial proof (or at least it did not in 2003, when I heard of it): Denote by $L(n)$ the set of all partitions of n into distinct parts with the s …