Simple representations of the Riemann function $ Xi $

The Riemann $ Xi $ function, defined as
$$
Xi (z) equiv – frac {1} {2} left (z ^ 2 + frac {1} {4} right) pi ^ { frac {1} {4} + i frac {z} {2}} Gamma left ( frac {1} {4} + i frac {z} {2} right) zeta left ( frac {1} {2} + iz right )
$$

has a number of beautiful properties. It's an entire function, unlike the $ Gamma $ and $ zeta $ the functions. His reflection formula $ Xi (-z) = Xi (z) $ is particularly easy to remember. The Riemann hypothesis for $ Xi (z) $ is also much simpler: all the zeros of $ Xi (z) $ are real.

On the other hand, this formula, in its definition, is very ugly and it is obvious that it is about all that the zeta function is shifted, rotated and scaled. Is there a better representation, possibly in the form of an integral function or other?

Nt.number theory – How to prove that the p-adic Galois representations attached to the Tate module of an abelian variety are from Rham directly?

Recently, I read a thesis on p-adic Galois representations and elliptic curves. Using the Tate Curve, the author has proved that the p-adic Galois representations associated with the Tate module of an elliptic curve is from Rham by constructing a base of $ D_ {dR} (V) $ explicitly in Page 53. Thus, the dimension of $ D_ {dR} (V) $ involves the representation is Rham.

Question: Is there also a standardization of an abelian variety on p-adic fields to prove that the representation attached to the Tate module is of Rham by building a base? And I think I can prove that the Hodge-Tate weights are 0 and 1 using a base.

By the way, I've heard that for a general variety $ V $, representations of $ G_K $ sure $ H ^ i_ {and} (V) $ is Hodge-Tate with weights 0, …, i-1. And I also want a reference for this fact.

Thanks in advance!

number theory – classifying 2-dimensional p-adic galois representations

Classification of one-dimensional representations, that is, characters $ G_K to E ^ times $, is a little more complicated than you suppose in your question. Such a character lands in $ O_E ^ times $ by compactness; and since $ O_E ^ times $ profinite, and class field theory identifies the abelianization of $ G_K $ with the professional completion of $ K ^ times $, we conclude that there is a canonical bijection between continuous characters $ G_K to O_E ^ times $ and continuous characters $ K ^ times to O_E ^ times $.

With regard to this bijection, we can read which characters are crystalline, semi-stable, of Rham or Hodge-Tate using the restriction of the character to $ O_K ^ times $: Details appear in Appendix B of B. Conrad, "Lifting Global Representations with Local Properties"; see also this question and this question.

As for $ n = $ 2, the best answer I can give is that for $ K = mathbf {Q} _p $, there is a bijection between two-dimensional representations of $ G _ { mathbf {Q} _p} $ and certain $ p $Banach representations in space $ GL_2 ( mathbf {Q} _p) $: it's the $ p $Ado-Langlands local correspondence of Colmez. You can then classify the 2-dimensional representatives that are Rham / crystalline according to their $ GL_2 $ representation. However, it is a bijection between a type of extremely complex object and another type of object also complicated; there is no hope of obtaining a realistic setting of all the representations involved. In addition, the extension of this correspondence to two-dimensional representatives of $ G_K $, for arbitrary K $, or to the three-dimensional representatives of $ G _ { mathbf {Q} _p} $, is an open problem despite a decade or more of very intensive efforts.

(Another point of view: if you just want to classify crystalline / semistable / Rham representatives, but you do not necessarily ask for a classification of all reps and how the crys / ss / dR sit inside, you can do it in some cases by sorting all the filters $ varphi $-modules, $ ( varphi, N) $-modules, etc. fulfilling the required conditions. See for example this paper of Dousmanis. But that does not tell you anything about what the non-Rham guys look like.)

reference request – Connections between linear representations and permutation representations

A finite group $ Gamma $ could be represented by a linear transformation

$$ rho: Gamma to mathrm {GL} ( Bbb R ^ d), $$

or by permutations

$$ phi: Gamma to mathrm {Sym} (n). $$

Of course, these can be interpreted as linear representations with matrices of permutations.

It seems to me that there are many interesting and non-trivial links between these two types of representations. In particular, one seems to be able to derive properties for one from them in the student from the point of view of the other.

Question: Is there a literature that explores these links in detail? What are the relevant search terms?

I am particularly interested in real representations, that is to say on $ Bbb R $. Here, I see many applications in geometry, for example: symmetrical polytopes, stiffness of symmetrical frames, etc.

Here are some examples of what I would consider interesting connections:

  • Can transitivity, primitivity, 2-closure or any other property of a group of permutations be well characterized in terms of decomposition into irreducible real linear representations?
  • A geometric property of an arrangement of symmetrical points (an orbiting polytope, if you will) can be well characterized by a property of the permutation group induced on points?

representation theory – Reference query: projective representations of a simple connected semi-simple Lie group elevator, in unitary representations

I recently became interested in the theory of quantum mechanics representations and read the following theorem:

Let $ G $ to be a Lie group simply connected with $ H ^ 2 ( mathfrak {g}, mathbb {R}) = 0 $ and let $ mathcal {H} $ to be a complex space of Hilbert. Then each projective representation $ rho: G to text {Aut} ( mathbb {P} ( mathcal {H})) $ lifts to a unitary representation $ pi: G to U ( mathcal {H}) $.

I am looking for a proof of the above theorem, does anyone have a reference where it is proved?

Regular, algebraic and cuspidal for automorphic representations

Given an automorphic representation of the general linear group on a totally real number field, it could have 3 properties: regular, algebraic (or "algebraic"?), And cuspidal. There is $ 2 ^ 3 = $ 8 possible combinations whose properties are true for a given representation. Can you give explicit examples for each of the 8 combinations?

Can you give examples for the general linear group on $ mathbb {Q} $? If yes, what is the minimum $ n $ (a sin $ mathrm {GL} _n $) so that all combinations appear?

reference request – $ PSL_2 ( mathbb {R}) $ representations of free groups

Let $ S_ {g, n} ^ b $ denote a gender surface $ g $ with $ n $ punctures and $ b $ boundary components. Suppose $ max {b, n } geq 1 $. It is obvious then that $ S_ {g, n} ^ b $ the deformation retracts into a bouquet of $ m: = 2g + n + b-1 $ circles and $ pi_1 (S_ {g, n} ^ b) $ is free on $ m $ generators.

Let $ m geq $ 2. Yes $ S_ {g, n} ^ b $ is equipped with a complete hyperbolic metric with a geodesic boundary, then it is known that there is a fuchsian group of the second type $ Gamma $, acting on the disc, such as $ S_ {g, n} ^ b $ can be rebuilt as $$ PSL_2 ( mathbb {R}) cup (S ^ 1- Lambda ( Gamma)) / Gamma $$ or $ Lambda ( Gamma) $ is the limit.

My question is this: given a faithful representation $ F_ {2g + n + b-1} to PSL_2 ( mathbb {R}) $, how to find the topological type of the corresponding surface? Notice, for example, $ F_4 $ could describe a surface with the kind $ 2 $ and $ 1 $ puncture or a kind surface $ 1 $ with $ 2 $ punctures and $ 1 $ limit component.

Ideally, there is a source or article where this type of question has been studied. (Otherwise, I will have to solve it myself).

Show that the characters of the $ phi_ {n} $ representations of $ SU (2) $ constitute a complete orthogonal set.

The question is given below:

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And the other questions mentioned are (I know the solutions of all):

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Could someone give me a hint for answers to the question?

Group Representations of $ C_5 $ – Mathematics Stack Exchange

If I have the cyclic group $ C_5 =$ and the regular left representation $ V = mathbb {C} C_5 $. Would the matrices of this representation (in the standard database) be defined by $ rho_g $, the 5×5 matrix with $ e ^ { frac {2 pi i} {5}} $ on the diagonal and zeros elsewhere? If not why?

opengl – Difference between two perspective projection representations

When looking for perspective projections on the Web, I encountered two different representations. One is not related to OpenGL and the other is strictly associated with it.

Projection 1

Projection 2

What is the relationship between these two projections? Both are called the same but the dice are quite different. The first seems to project on the xy (z = 0) plane, that of OpenGL on the z = close plane.