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2021-12-10

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Given this natural precedent, one goal for is to take the proteins themselves as building modules and synthesize artificial regulatory circuits that have preselected inputs and outputs. As before, the motivation is not so much to construct 'toys', but rather to use the challenge of a synthetic goal to discover principles that connect the chemistry of signal transduction to emergent regulatory properties in complex biology.

**Idempotents and Structure of Simple GL(n,p) modules in the describing characteristic**

It's probably safe to say that beyond $n=2$ (and possibly $n=3$), little is known about the representations of these groups in characteristic 2. While there is a lot of general theory aimed at organizing such things for all reductive groups and at least most primes, the small primes pose extra problems at every step. On the other hand, the general theory (as in Jantzen's foundational book Representations of Algebraic Groups) tends to start over an algebraically closed field, then eventually specialize to smaller fields including finite ones. For small $n$ and $p$ there are some scattered computations in the literature but little in the way of a pattern. Indeed, for a prime like 2 this overlaps with the difficult problem of describing modular representations of symmetric groups. While your results may be effective (using something like brute force), it's not easy to reconcile concrete methods including computer use with the urge to find theoretical patterns. Still, it's good to investigate the questions by whatever means are possible, in the absence of general answers.ADDED: I tried to summarize what was known at the time (at least in the algebraic group literature) in Chapter 19 of my LMS Lecture Note Series 326 Modular Representations of Finite Groups of Lie Type (2006). I was aware of some work by algebraic topologists as well as the long tradition involving Dickson invariants and some more recent ideas such as Steenrod operations. But I may have overlooked some relevant papers. Steinberg's Lie-theoretic classification provides at least an outline of what is possible over finite fields, in terms of highest weights. But the details are tricky to fill in.

**Coproducts of modules over an algebraic monad**

I think the point is that an operation in an algebraic theory (even a noncommutative one) need not preserve the order of its inputs. There is a binary operation in the theory of groups which takes the input $(g,h)$ to the product $h g$. More generally, the symmetric group on $n$ letters acts on the set of $n$-ary operations, although in general the action may not have many fixed points. Thus, you can always find an operation which permutes all the $M$-inputs to the front and all the $N$-inputs to the rear

**On Similarities and Differences Between Right and Left Modules over a Ring**

The "external multiplication" of scalars from a ring $R$ is modelled (if tacitly) on the internal multiplication of scalars within the ring. I will discuss two angles of evidence for this viewpoint.Semisimplicity. Consider semisimple $R$-modules $M$. Then $M$ decomposes as a direct product of simple modules as $Mcong M_1timescdotstimes M_n$. Simple modules are characterized as being quotients of the scalar ring ($Ncong R/I$) up to isomorphism. This tells us very directly that the scalar operation is modeled very closely off of the multiplication operation within $R$. More generally modules can be more complicated than semisimple ones, but "more complicated" is no reason to expect the ring's multiplication operation to stop having control or influence over the module structure.Action maps. An $R$-module $M$ is essentially encoded as a homomorphism $Rto

m End(M)$. Thus the scalar operation is encoded by a homomorphic image of $R$, which is obviously dependent on the nature of $R$'s internal operations.In the above I do not mention left versus right; fill in the blanks. That multiplication in $R$ is different between left v. right perspectives means the ideal theory may differ as well.Categorical thoughts. If $R$ is commutative then all ideals are two-sided automatically, and any left $R$-module can be turned canonically into a right $R$-module (simply define the same action!). From a high-brow viewpoint with the right language we can say there is a natural equivalence between the categories of left $R$-modules and right $R$-modules. However in the noncommutative setting the most we can claim is an equivalence between left $R$-modules and right $R^

m op$-modules, where $R^

m op$ stands for the opposite ring. In general $R

otcong R^

m op$ (tough examples here).

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