Fix a finite dimensional graded vector space $V$, a differential $d$ on $V \otimes V $ $(i.e.\,\, d^2=0\,\, and \,\, deg\, d =1)$ such that $(d_{1,2} \otimes id_3)$ anti-commutes with $((-1)^{deg}_1 \otimes d_{2,3}) $ on $V \otimes V \otimes V$. Define $V_N$ to be cohomology of $(d_{1,2}+d_{2,3}+d_{3,4}+...+d_{n-1,n})$ on $V^{\otimes n}$. Here $d_{i,i+1}$ means $d$ acting on i-th and (i+1)-th factor (and it’s (-1)^{deg} on factors before i and id on factors after (i +1) ) . Can we prove that $dim\, (V_N)\sim Ca^NN^b $for some algebraic number a and integer b? (Actually I guess it’s a combination of polynomials and exponentials)
12
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3$\begingroup$ How do you build a complex from $V\otimes V$? How do you define $d_{i,j}$? $\endgroup$– foscoCommented Jan 29, 2021 at 10:00
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$\begingroup$ The underlying space of the complex (if there is a compatible grading) is $V^{\otimes n}$, where $d_{ij}$ means $d$ acting on i-th and j-th factor (and it’s id on other factors) $\endgroup$– Xu KaiCommented Jan 29, 2021 at 14:42
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1$\begingroup$ Are you sure this is a differential? If $n=4$ I get $(d_{1,2}+d_{2,3} + d_{3,4} + d_{4,1} )^2 = d_{1,2}d_{3,4} + d_{3,4} d_{1,2} + d_{2,3}d_{4,1}+ d_{4,1} d_{2,3}= 2 d_{1,2}d_{3,4} + 2 d_{2,3} d_{4,1}$ which seems nonzero. $\endgroup$– Will SawinCommented Jan 29, 2021 at 14:49
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$\begingroup$ Oh sorry this is my mistake, I really want V to be graded and impose Koszul sign rule so that $d_{12}$ anticommutes with $d_{34}$. I will correct the question $\endgroup$– Xu KaiCommented Jan 29, 2021 at 14:53
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$\begingroup$ Many thanks for point this out! $\endgroup$– Xu KaiCommented Jan 29, 2021 at 15:01
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