# Simplicial set of permutations

Let $$S_k$$ be the set of all permutations of $$k+1$$ elements $$0,1,...,k$$. introduce boundary maps $$d_i : S_k \rightarrow S_{k-1}$$ by deleting from permutation $$\eta$$ element $$\eta(i)$$ and monotone reordering and degeneracy
$$s_i :S_k \rightarrow S_{k+1}$$ by adding 1 to all elements with $$\eta(j)>\eta(i)$$ and incerting into the result a new element $$\eta(i)+1$$ right after $$\eta(i)$$ on $$i+1$$ place. It is a simplicial set, contractible and classifies reorderings of simplicial sets.

Is it known? May be in higher symmetric something?

(Update) Boris Tsygan pointed the right direction in Facebook duscussion

The object is classical and it has a name "Symmetric crossed simplicial group”. It was introduced almost simultaneously in

Appendix A10, page 191 “Symmetric objects” B. L. Feigin and B. L. Tsygan “Additive K-theory” 1987 K-theory, arithmetic and geometry, Semin., Moscow Univ. 1984-86 LNM 1289

Krasauskas, R. "Skew-simplicial groups", Lithuanian Mathematical Journal, Jan 1987 vol 27 issue 1 p. 47--54

And independently

Zbigniew Fiedorowicz and Jean-Louis Loday “Crossed simplicial groups and their associated homology” Trans. Amer. Math. Soc. 326 (1991), 57-87

It has big value in everything symmetric. Geometric realization $$|S_\bullet|$$ is the topological group structure on infinite dimensional sphere.

• In $s_i$ you mean that we insert something like $\eta(i)+1/2$? $\eta(i)+1$ may be already in the permutation. – Fedor Petrov Jul 10 '19 at 16:33
• $s_3(51243)=612354$, right? – Fedor Petrov Jul 10 '19 at 16:55
• @FedorPetrov right. Sorry was fighting with phone replying. Snd finnsly killed the comment. – Nikolai Mnev Jul 10 '19 at 17:01
• @NikolaiMnev: If it's $\eta(i)+1$, then what about Fedor's comment? There is no canonical monotone reordering when two entries are equal. – darij grinberg Jul 11 '19 at 12:46
• @NikolaiMnev : Not everyone on MathOverflow is on Facebook. (I am not on Facebook.) I would recommend that you summarize the conclusions of the Facebook discussion here, either by editing your question or by posting an answer to your own question. – Timothy Chow Jul 12 '19 at 16:06

I think something equivalent (or at least closely related) to this has been studied in the combinatorics literature.

A CW complex of course has a poset of faces. In this case, this poset is obtained by ordering permutations by subword inclusion up to deletion and monotone reordering. The keyword used in the combinatorics literature for this sort of subword inclusion is permutation patterns.

Now, if a CW complex is regular, then the order complex of the face poset is homeomorphic to the complex. As you point out in the comments, the simplicial set has $$n!$$ faces of dimension $$n$$, and in particular has a single vertex. So the simplicial set isn't regular, as I'd initially thought it might be, and your question doesn't reduce directly to this poset. It certainly seems like the two objects should be closely related, however.

In any case, the lattice of permutations ordered by pattern containment has been studied by Jason Smith. See, for example, the paper

Smith, Jason P., A formula for the Möbius function of the permutation poset based on a topological decomposition, Adv. Appl. Math. 91, 98-114 (2017). ZBL1370.05227.

That paper cites also his earlier papers on the topic.

• Thank you for the comment. It is certainly related to permutation patterns. I can simply count nodegenerate simples here as something like subfactorial, for example, using pattern views. But itself it is very very far from regular complex. it has a single vertex two adges and factorials of other simplexes glued on this tiny 8. may be it is glued from some classic regular complexes in an interesting singular way. Itbwold be cool to know. thank you for the reference! – Nikolai Mnev Jul 11 '19 at 8:59
• Yes, I should've caught the one vertex thing. Edited to better reflect. I'm not familiar enough with the simplicial set way of thinking to know if there's some other well-travelled path between the two objects. – Russ Woodroofe Jul 11 '19 at 10:59
• @RussWoodfore Geometric realization of this thing looks strange in finite parts but actually an infinite dimensional Lie group as a whole thing. Wonders of simplicial topology. I still surprised – Nikolai Mnev Jul 14 '19 at 9:37