Representation theory of 3-stage nilpotent finite groups

I am interested in understanding the representation theory of certain finite nilpotent groups (via the complex numbers). The groups $$G$$ of interest have the following properties:

1) G is $$3$$step nilpotent finite group

2) G is a $$2$$-Group

3) G is an extension of a 2-stage nilpotent group H by $$mathbb {Z} / 2 mathbb {Z}$$,

I know that the irreducible representations of a finite nilpotent group are monomial, and there is a unified construction of these representations in the case of $$2$$step nilpotent groups. For a $$2$$step nilpotent group $$H$$, any irreducible representation $$rho$$ of $$H$$ the dimension greater than $$1$$ is induced from a character of a fixed maximum Abelian subgroup $$S$$ of $$H$$, Actually $$rho$$ is determined by its central character.

Question:

What can be said about the representation theory of $$G$$? Can we identify a minimal set of subgroups that can be used to construct each irreducible representation as a monomial representation? (For example for a $$2$$step nilpotent group We need a maximum Abelian subgroup and the group itself.)

I would appreciate thoughts on this question and references to 1-3 above. That is, all references to the representation theory of $$3$$-step nilpotent groups, $$2$$-Groups or double extensions of $$2$$step nilpotent group or any combination thereof.

Sincerely yours,

Moshe

Complexity Theory – Is there a term like "effectively calculable reductions" or would this not make sense?

Most of the reductions in NP hardness evidence I've come across are effective in the sense that given the instance of our difficult problem, they provide a polynomial time algorithm for our problem using reductions. All reductions of the 21 classic problems that R. Karp considers work in this way. A very simple example could be reducing INDEPENDENT_SET to CLIQUE. Simply create the complement diagram of your input.

However, when you consider the proof of the famous Cook-Levin theorem that SAT is NP-complete, the reduction assumes a non-deterministic TM and a polynomial that by definition exists. But for me it is not clear how to get this polynomial effectively, which means that given a nondeterministic TM that I know is in polynomial time, it is not clear how to compute this polynomial, and I suspect very much that it is unpredictable at all.

Just because of the encoding of NP-complete problems by the class of non-deterministic polynomial time TMs (the polynomial itself is not encoded to my knowledge) I don't see any way to effectively reduce the above evidence just shows that there are some but not how to get it.

Perhaps I have misunderstood something, but I have the impression that the reductions given are usually stronger in the sense that they are actually predictable, i.e. H. If there is a problem, we can calculate our reduction and not just know its existence.

Has this ever been noticed? And if so, is there a term like "effectively calculable reduction", or would it be impractical to limit the reduction so that it is predictable? From a practical point of view and also from the way I sometimes see reductions introduced ("as if we have an algorithm to convert one instance to another"), it would be very desirable to know how to actually use this algorithm / reduction it seems more natural to request it, but why isn't it done?

Complexity theory – NP hard or not: two partitions with finite irrational input

Given a sentence $$N$$ with n + 2 numbers,

The first n numbers are positive and rational, the sum of which is 1

The n + 1st number is $$sqrt {2}$$

The n + 2nd number is $$2- sqrt {2}$$

Determine if we can find a subset of N so that the sum of the subset is 3/2.

I come across this problem from the referee who has been arguing it ever since $$sqrt {2}$$ is irrational, I shouldn't reduce it from a two partition problem.

May I have your comments and a reference book so I can go into the fact that this is still a NP-difficult problem.

Set theory – permutation numbers \$ mathfrak j_1 \$, \$ mathfrak j_2 \$, they are equal to \$ mathrm {non} ( mathcal M) \$

Let us consider the following main features of the continuum:

$$mathfrak j_1: = min {| H |: H subset S_ omega ; wedge ; forall A in ( omega) ^ omega ; exists h in H$$ so that $$h (A) cap A$$ is infinite$$}$$;;

$$mathfrak j_2: = min {| H |: H subset S_ omega ; wedge ; forall A, B in ( omega) ^ omega ; exists h in H$$ so that $$h (A) cap B$$ is infinite$$}$$,

Here $$S_ omega$$ is the permutation group of the set $$omega$$, and $$( omega) ^ omega$$ is the family of infinite subsets of $$omega$$,

It can be shown that $$max { mathfrak b, mathfrak s } le mathfrak j_1 le mathfrak j_2 le mathrm {non} ( mathcal M)$$,

Problem 1. is $$mathfrak j_1 = mathfrak j_2$$ in ZFC?

Problem 2. is $$mathfrak j_2 = mathrm {non} ( mathcal M)$$ is ZFC?

Annotation. It can be shown that the cardinals $$mathfrak j_ {1}, mathfrak j_2$$ are each the same as the cardinals $$mathfrak j_ {2: 1}, mathfrak j_ {2: 2}$$ defined in this MO problem). Therefore problems 1 and 2 correspond to problems 0 and 1 from this MO contribution. But perhaps the corresponding descriptions of these cardinals help somehow.

Does the theory of the cubic type still match the univalent excluded middle and univalent choice?

I would like to formalize some bachelor's mathematics studies in cubic agda and learn cubic type theory. The problem is that I need a one-value excluded medium and one-value selection (and possibly a change in sentence size). I know that they agree with homotopy type theory (although the calculation is lost when using the axiom), but this type theory is more cubic (in the sense that univalence is a theorem). Is this axiom still consistent in the cubic environment? Is there a better way to create classic theorems in cubic type theory?

nt.number theory – Unexpected probability with √2 and parity

This article focuses on a very specific part of this long article. Consider the following map:
$$f: n mapsto left { begin {array} {ll} left lfloor {n / sqrt {2}} right rfloor & text {if} n text {even,} \ left lfloor {n sqrt {2}} right rfloor & text {if} n text {odd.} end {array} right.$$

To let $$f ^ {r + 1}: = f circ for ^ r$$Consider the orbit of $$n = 73$$ on the iterations of $$f$$i.e. the sequence $$f ^ r (73)$$:: $$73, 103, 145, 205, 289, 408, 288, 203, 287, 405, 572, 404, 285, 403, 569, 804, 568, 401, dots$$
Now consider the probability of $$m$$The first terms of the sequence $$f ^ r (73)$$ be straight: $$p_ {0} (m): = frac {| {r
Then $$p_1 (m): = 1-p_0 (m)$$ is the probability of $$m$$The first terms of office $$f ^ r (73)$$ be strange.

If we calculate the values ​​of $$p_i (m)$$ to the $$m = 10 ^ { ell}$$. $$ell = 1, dots, 5$$We get something unexpectedly::
$$scriptsize { begin {array} {c | c} ell & p_0 (10 ^ { ell}) & p_1 (10 ^ { ell}) newline hline 1 & 0.2 & 0.8 newline hline 2 & 0.45 & 0.55 newline hline 3 & 0.467 & 0.533 newline hline 4 & 0.4700 & 0.5300 newline hline 5 & ​​0.46410 & 0.53590 end {array}}$$
It appears that $$p_0 (m)$$ does Not converge to $$1/2$$, but that $$p_0 (m) sim 46.5 %$$ and $$p_1 (m) sim 53.5 %$$

question: Is it true that $$p_0 (m)$$ does not converge $$1/2$$?
Suppose it converges, let it $$alpha$$ be his limit. is $$alpha$$ conclude to $$0.465$$? What is the exact value?

The following picture shows the values ​​of $$p_0 (m)$$ to the $$100 ::

Note that this phenomenon is not specific to $$n = 73$$, but seems to happen as often as $$n$$ is large, and then the analog probability seems to converge to it $$alpha$$, If $$n <100$$, then it happens for $$n = 73$$ only but for $$n < 200$$it happens for $$n = 73, 103, 104, 105, 107, 141, 145, 146, 147, 148, 149, 151, 152, 153, 155, 161, 175, 199$$, And for $$10000 le n < 11000$$it happens exactly $$954$$ Ones.

Below is the picture as above, but for $$n = 123456789$$::

Alternative question: Is it true that the amount of $$n$$ For which phenomenon does natural density one occur? Is it cofinite? If it happens, it is the same constant $$alpha$$?

p adic number theory – conjugation of maximal algebraic tori

Accept $$G$$ is a connected, reductive algebraic group over a non-Archimedean local field $$F$$which is divided over a finite extent $$E / F$$,

I often see a result that says "everything is maximum $$F$$-Tori are conjugated over $$E$$", by which I understand the following: Let $$G (E)$$ denote the $$E$$-Dots of the algebraic group $$G$$;; then for each maximum $$F$$-tori $$T, T$$ of $$G$$is there $$x in G (E)$$ so that $$T (E) = xT & # 39; (E) x ^ {- 1}$$,

In addition, the definitions show that if $$T, T$$ are maximum $$F$$-tori from $$G$$then there is an isomorphism of $$T (F)$$ on to $$T & # 39; (F)$$ which is defined via $$E$$,

My question is: Can the isomorphism be assumed to be conjugation in the second statement (as in the first statement)? That means: it follows from these results that if $$T, T$$ are maximum $$F$$-tori in $$G$$then it exists $$x in G (E)$$ so that $$T (F) = xT & # 39; (F) x ^ {- 1}$$?

Any help (including proof of the first statement) is greatly appreciated!

Complexity Theory – Logspace Computable of the composition

The Bitgraph of $$f: {0,1 } ^ * rightarrow {0,1 } ^ *$$ is the language:

$$text {BIT} _f: = { : 1 leq i leq | f (x) | text {and the i-th bit of} f (x) text {is} 1 }$$

It was said that $$f$$ is Logspace predictable if $$text {BIT} _f$$ is decidable in space $$O (log (n))$$, Decidable means that there is a Turing machine $$M$$ so that:

1. if $$in text {BIT} _f$$ then $$M () = 1$$
2. if $$notin text {BIT} _f$$ then $$M () = 0$$

Prove that the composition $$(f circ g) (x) = f (g (x))$$ of two logspace calculable functions $$f, g$$ is also a logspace predictable function.

Any advice on this exercise? What I've tried so far is to play with the composition of the two Turing machines $$f$$ and $$g$$, but I was unsuccessful because there was always a case analysis.

gr.group theory – embedding a group in a simple group

It is known that each group G can be embedded in a simple group S, see e.g. B. The discussion under Can each group be embedded in a simple group?

My question is whether one can get an embedding so that the surrounding group S has commutator width 1 (i.e. each element of S can be represented as a single commutator).

nt.number theory – solutions in prime numbers of the equation \$ , 3p ^ 2 + q ^ 2 = r ^ 2 + 3 \$

Let us consider the diophantine equation $$, 3p ^ 2 + q ^ 2 = r ^ 2 + 3$$,

Actually, I'm only interested in tern solutions $$, (p, q, r) ,$$ of prime numbers.

It is easy to prove that if $$, (p, q) ,$$ is a pair of twin prime numbers $$, (p lt q)$$, then $$, (p, q) ,$$ solves the equation if $$, p ,$$ is a Sophie Germain Prime and $$, r ,$$ represents its safe prime number (that is $$, r = 2p + 1$$).

Likewise if $$, (p, q) ,$$ is a pair of twin primes with $$, p gt q$$, then $$, (p, q) ,$$ solves the equation if $$, r = 2p-1$$,

So here are some solutions to the given equation:

$$(3,5,7) ; ; (5,7,11) ; ; (11,13,23) ; ; (29,31,59) ; ; (41.43, 83) ; ; …$$

$$(7,5,13) ; ; (19,17,37) ; ; (31,29,61) ; ; …$$

We force to find other classes of solutions $$, r = p + 2q$$, This restriction means that only the pairs are taken into account $$, (p, q) ,$$ meet the following condition:

$$5 (q ^ 2 + 1) = 2 ((p-q) ^ 2 + 1) ; ; ; ; ; ; ; ; ; (*)$$

A tern satisfactory $$, (*) ,$$ is $$, (13,5,23)$$,

I ask you to find other, possibly more general, classes of solutions to the given equation.