Is There a Mistake in This CLQG Thesis?

[julian]

I've been reading this interesting thesis recently posted. But I think I spotted a mistake. On page 52 of "Semi-Classical Holomorphic Transition Amplitudes in Covariant Loop Quantum Gravity":

https://arxiv.org/pdf/2001.04651.pdf

It says "In general, removing $k$ vertices from $[v_0, v_1, . . . , v_p]$ leaves us with a $p − k$-simplex contained in the original $p-$simplex. As there are $\binom{p+1}{k+1}$ possibilities to remove $k$ elements from a list of length $p + 1$ we just proved..."

This is a mistake, it should say there are $\binom{p+1}{k}$ possibilities!

Or if you like there are $\binom{p+1}{p-k+1}$ possibilities.

If you put $n=p-k$ you get the same result as stated on page 67 of Nakahara, "Geometry, Topology and Physics" which says (adjusting for notation) that the number of $n-$faces in a $p-$simplex is $\binom{p+1}{n+1}$.

Yep?

 

[julian]

Thanks for the like @atyy.

I think maybe the author needs to change Lemma 3.2.1 in https://arxiv.org/pdf/2001.04651.pdf to:

"The number of $n-$simplices (where $n=p-k$) contained in a $p-$simplex, which we denote as $N^p_n$, is given by

$$
N^p_n = \binom{p+1}{n+1} \quad \text{for } 0 \leq n \leq p .
$$"

because then the formula that he uses for the Euler characteristic in Eq (3.28), namely $\chi (\sigma^{(d)}) = \sum_{n=0}^d (-1)^n N^d_n$, would be correct (i.e. in accordance with page 86 of Nakahara).Eq (3.28) appears to be the only place Lemma 3.2.1 is applied, and so the mistake he made has no impact on any of the results of the thesis.

It appears to be a very nicely written thesis.

 

[atyy]

Maybe you could email the author? And report back to us what he says

 

[julian]

Maybe you could email the author? And report back to us what he says

I did email him, but didn't receive a response - but I then I realized that is not his current email address...

I actually know somebody who knows him, I might email them at some point.