I wonder what kinds of closed form infinite products of matrices, elements of Banach algebras, and complex numbers arise from the rank-into-rank embeddings.

Suppose that $\lambda$ is a cardinal and $j_{1},\dots,j_{k}:V_{\lambda}\rightarrow V_{\lambda}$ are non-trivial elementary embeddings. Let $\mathrm{crit}_{n}(j_{1},\dots,j_{k})$ be the $n$-th element in the set $\{\mathrm{crit}(j)\mid j\in\langle j_{1},\dots,j_{k}\rangle\}$. As a convention, we shall assume implied parentheses are grouped on the left, so $j*k*l=(j*k)*l$. For more information about the algebraic operation $*$ that arises from the rank-into-rank embeddings, please read Chapter 11 in the Handbook of Set Theory.

Let $p_{n,j_{1},\dots,j_{k}}(x_{1},\dots,x_{k})$ denote the non-commutative polynomial defined $$1+\sum\{x_{a_{1}}\dots x_{a_{s}}\mid\mathrm{crit}(j_{a_{1}}*\dots*j_{a_{s}}) =\mathrm{crit}_{n}(j_{1},\dots,j_{k}),$$ $$\mathrm{crit}(j_{a_{1}}*\dots*j_{a_{r}})<\mathrm{crit}_{n}(j_{1},\dots,j_{k})\,\text{for all}\,1\leq r<s\}.$$

If the elementary embeddings $j_{1},\dots,j_{k}$ are unambiguous, then we shall write $p_{n}(x_{1},\dots,x_{k})$ for $p_{n,j_{1},\dots,j_{k}}(x_{1},\dots,x_{k})$.

The variables $x_{1},\dots,x_{k}$ do not commute with each other (so $x_{1},\dots,x_{k}$ should be thought of as matrices of elements of a Banach algebra).

The polynomials $p_{n}(x_{1},\dots,x_{k})$ satisfy the infinite product formula $$\lim_{n\rightarrow\infty}p_{n}(x_{1},\dots,x_{k})\cdot\dots\cdot p_{0}(x_{1},\dots,x_{k})=\frac{1}{1-(x_{1}+\dots+x_{k})}.$$

For example, if $j_{1}=\dots=j_{k}$, then $p_{n}(x_{1},\dots,x_{k})=1+(x_{1}+\dots+x_{k})^{2^{n}}$ for all $n\in\omega.$

Can anyone give a non-trivial example of a sequence of distinct non-trivial elementary embeddings $j_{1},\dots,j_{k}$ along with $r\times r$ matrices $A_{1},\dots,A_{k}$ where if $B_{n}=p_{n}(A_{1},\dots,A_{k}),$ then

$1-(A_{1}+\dots+A_{k})$ is non-singular,

there is a closed form expression for the sequence $(B_{n})_{n\in\omega}$ (in particular, if each entry in each $A_{i}$ is algebraic over $\mathbb{Q}$, then the coefficients in $B_{n}$ should be computable in polynomial time at the very least),

$$\lim_{n\rightarrow\infty}B_{n}\cdot\dots\cdot B_{0}=\frac{1}{1-(A_{1}+\dots+A_{k})},$$

If $\alpha=\mathrm{crit}(j_{i})$ for some $i$, then $$\sum\{A_{i}\mid 1\leq i\leq k,\mathrm{crit}(j_{i})=\alpha\}$$ is non-singular, and

The sequence $(B_{n})_{n\in\omega}$ is not eventually identically $1.$

I hope that conditions 4 and 5 rule out all the trivial cases.

I am also interested in a few generalizations of this question and will appreciate answers in the generalized questions. For example, one way to generalize polynomials is to use infinitely many variables $x_{r}$ corresponding to infinitely many elementary embeddings. Another generalization would be to choose $A_{1},\dots,A_{k}$ from some Banach algebra (or some other space) or for $A_{1},\dots,A_{k}$ to simply be complex numbers. Another way to generalize this question is to use algebraic structures that resemble rank-into-rank embeddings in the sense of having critical points, a composition operation, etc but which do actually arise in the algebras of elementary embeddings.

Some of my computer calculations indicate that there could be these non-trivial infinite products including the following candidate for an answer to this question.

Suppose that $k=2$ and $j:V_{\lambda}\rightarrow V_{\lambda}$. Let $j_{1}=j*j,j_{2}=j*j*j$. Then the following sequence is the sequence $(p_{0}(x,ix),\dots,p_{15}(x,ix))$:

$( i\cdot x+1, x+1, 1, i\cdot x^2+1, -x^4-x^3+1, i\cdot x^3+1, x^7+1, (-i)\cdot x^6+1, 1, 1, 1, 1, 1, 1, 1, 1 )$.

The value of the polynomial $p_{16}(x,ix)$ is unknown, and I do not know if the sequence of polynomials $(p_{n}(x,ix))_{n}$ is in any closed form or can be computed in polynomial of $n$ time. The printout for $(p_{0}(x,y),\dots,p_{15}(x,y))$ is 33982 characters long, so the niceness of the polynomials $(p_{0}(x,ix),\dots,p_{15}(x,ix))$ is quite unusual.

We also have $(p_{0,j,j*j}(i,1),\dots,p_{11,j,j*j}(i,1))=(1+i, 1, 1+i, 1, 0, 1, 1, 1, 1, 1, 1, 1),$ and

$(p_{0,j*j,j*(j*j)*j}(1,i),\dots,p_{15,j*j,j*(j*j)*j}(1,i))= (1+i, 1, 1, 1+i, 1+i, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 )$

One should take caution and for the Laver tables, one should never assume that any pattern continues since the Laver tables are filled with temporary patterns and even some long lasting patterns which must terminate under large cardinal hypotheses.