I've got a problem related to $(p,q)$-shuffles that comes from the Eilenberg-Zilber map $\nabla$ when I tried to show that this map is associative in the sense that $\nabla(\nabla\otimes 1)=\nabla(1\otimes \nabla)$.

On one side, I have for $(p,q)$-shuffles $(\mu,\nu)$ and $(p+q,r)$-shuffles $(\alpha,\beta)$ the expression

$$\sum_{(\alpha,\beta), (\mu,\nu)} sgn(\alpha,\beta)sgn(\mu,\nu)s_\beta(s_\nu(a))\otimes s_\beta(s_\mu(b))\otimes s_\alpha(c)$$

and on the other side, for $(q,r)$-shuffles $(\delta,\gamma)$ and $(p,q+r)$-shuffles $(\xi,\omega)$

$$\sum_{(\xi,\omega),(\delta,\gamma)} sgn(\xi,\omega)sgn(\delta,\gamma)s_\omega(a)\otimes s_\xi(s_\gamma(b))\otimes s_\xi(s_\delta(c))$$

In order to show that the latter expression is the same as the former, I would like to combine $(\xi,\omega)$ and $(\delta,\gamma)$ to obtain a $(p,q)$-shuffle and a $(p+q,r)$-shuffle such that the products of its signs is precisely $sgn(\xi,\omega)sgn(\delta,\gamma)$.

Sice $(\xi,\omega)$ is a $(p,q+r)$-shuffle, I can just take $(\xi_1,\dots, \xi_p,\omega_1,\dots,\omega_q)$ as $(p,q)$-shuffle. But then, to obtain a $(p+q,r)$-shuffle I have to permute the $\xi_i$ and the $\omega_j$ to order them in a way that the sign is the one I want. Somehow I should use $(\delta,\gamma)$ to do that, but this is a $(q,r)$-shuffle, so I don't know what to do.

  • $\begingroup$ The tags (shuffles) and (chain-complexes) currently don't have any user guidance, which leads to a higher chance they being removed by other users. As the tag creator, would you mind to add some? $\endgroup$ – YuiTo Cheng Apr 21 at 14:19
  • $\begingroup$ @YuiToCheng I don't have tag wiki edit privileges so I cannot do that. $\endgroup$ – Javi Apr 21 at 14:29
  • $\begingroup$ Me neither. Still, you can suggest edits. $\endgroup$ – YuiTo Cheng Apr 21 at 14:30
  • $\begingroup$ @YuiToCheng ok I will $\endgroup$ – Javi Apr 21 at 14:30

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