rt.representation theory – Question on geometric lemma in the $p$-adic group representation

To check whether my understaning on the geometric lemma is right, I would like to ask some specific question related to it.

Let $F$ be a $p$-adic local field of characteristic zero. Let $W_n$ be a symplectic space over $F$ of dimension $2n$. Let $Sp_{n}(W_n)$ be a symplectic group and $B_n$ its standard Borel subgroup. Let $Q_{t,n-t}$ be a standard parabolic subgroup of $G$ preserving a totally isotropic subspace of $W_n$ of dimension $t$.

Let ${chi_i}_{1 le i le n}$ be unramified characters of $GL_1(F)$ such that $chi_i(omega)=1$ and let $pi$ be the normalized parabolic induced representation $text{Ind}_{Q_{3,n-3}}^{Sp_{n}}(chi_1 circ text{det}_{GL_3},chi_2,cdots,chi_{n-2})$.

I am considering $J_{Q_{2,n-2}}(pi)$, the normalized Jacquet module of $pi$ to $Q_{2,n-2}$. By the geometric lemma, there is some filtration of $J_{Q_{2,n-2}}(pi)$

$$0=tau_0 subset tau_1 subset cdots subset tau_m=J_{Q_2}(pi)$$ such that $tau_{i}backslashtau_{i+1}$ is some induced representation of the Jacquet module of the inducing data of $pi$.

I guess that such subquotient appearing in this filtration has the form $|cdot|^{-frac{1}{2}}cdot(chi_1 circ text{det}_{GL_2}) boxtimes text{Ind}_{B_{n-2}}^{Sp_{n-2}}(chi_1′,chi_2′,cdots,chi_{n-2}’)$ or
$text{Ind}_{B_2}^{GL_2}(chi_1′,chi_2′) boxtimes text{Ind}_{Q_{3,n-5}}^{Sp_{n-2}}(chi_1 circ text{det}_{GL_3},chi_4′,cdots,chi_{n-2}’)$?

(Here, ${chi_1′,cdots,chi_{n-2}’} subset {chi_1,cdots,chi_{n-2},chi_1^{-1},cdots,chi_{n-2}^{-1}}.)$

Any comments are highly appreaciated!

java – The origin server did not find a current representation for the target resource or is not willing to disclose that one exists

I am attempting to run a curl request on my java web service project that should return XML code to the server, via the command prompt however I am running into the above error. I’ve tried many different techniques, and it seems to me that for some reason the server is unable to find the file, however I’ve included a screenshot showing that the predictions.jsp file is in the src file where my cwd is pointing in command line. if anyone is able to help me that would be greatly appreciated as I am new to web services in general and have no idea where I should go from here. Please let me know if you need any additional information to help answer my question and I will gladly include it. I’ve included the screenshots from my command line to show the complete error message as well as the cwd to show that the file predictions.jsp is included in the directory.

image of current working directory

image of full command line error

This project was originally developed using Eclipse IDE however running the project using the Apache Tomcat Server packaged with my Eclipse Install does not work either, as I simply get a blank screen when running the predictions.jsp file from that server.

representation theory – Completely reducible Lie module implies existence of a complementar module

The following is the exercise 6.2 in Humphreys’ Introduction to Lie Algebra and Representation Theory. I wish to know if my attempt of the ‘only if’ direction is correct.

exercise

Suppose $V$ is a direct sum of irreducible sub-modules $V_1,dots,V_n$. Take a submodule $W$ of $V$, we have that $W cap V_i$ is a submodule of $V_i$ hence $V_i$ is $0$ or $V_i$. Take $I={i: V_i cap W=0 }$ and define $W’$ as

$$W’=bigoplus_{i in I} V_i,$$

this subspace does the job: we have direct sum $V=W oplus W’$ and that $W’$ is a submodule of $V$.

I’m asking this question because every other argument I have found on the web seems a bit dissimilar, so I think I’m missing something.

proof

fa.functional analysis – Representation of an arbitrary element on a fermionic Fock Space

Let $mathcal{H}$ be a Hilbert space with orthonormal basis ${varphi_{k}}_{kin I}$. Take $mathcal{H}^{otimes n} := overbrace{mathcal{H}otimescdotsotimes mathcal{H}}^{mbox{$n$ times}}$. An element of $mathcal{H}^{otimes n}$ can be expressed as:
$$psi = sum_{{k_{1},…,k_{n}}subset I}alpha_{k_{1},…,k_{n}}(varphi_{k_{1}}otimes cdots otimes varphi_{k_{n}})$$
with $alpha_{k_{1},…,k_{n}} = langle varphi_{k_{1}}otimescdotsotimesvarphi_{k_{n}},psirangle$. Let us define $sigma^{*}$ as an operator on $mathcal{H}^{otimes n}$ which acts on the basis elements as:
$$sigma^{*}(varphi_{k_{1}}otimes cdots otimes varphi_{k_{n}}) := varphi_{k_{sigma(1)}}otimescdots otimes varphi_{k_{sigma(n)}}$$
where $sigma$ is a permutation of the set ${1,…,n}$. We extend $sigma^{*}$ to all $mathcal{H}^{otimes n}$ by linearity. Now, one can define:
$$A_{n}:= frac{1}{n!}sum_{sigma}epsilon_{sigma}sigma^{*}$$
an antisymmetrization operator on $mathcal{H}^{otimes n}$. Here $epsilon_{sigma}$ is the sign of the associate permutation $sigma$. Then $A_{n}$ is an orthogonal projection and, if $A_{n}mathcal{H}^{otimes n}$ denotes its range, the fermionic Fock space is defined to be:
$$mathcal{F}_{f}(mathcal{H}) := bigoplus_{n=0}^{infty}A_{n}mathcal{H}^{otimes n}$$
with $A_{0}mathcal{H}^{0} := mathbb{C}$.

Alternatively, let us say that a tensor $psi in mathcal{H}^{otimes n}$ is antisymmetric if $sigma^{*}psi = epsilon_{sigma}psi$ for every permutation $sigma$. Take $wedge^{n}mathcal{H}$ to be the subspace of all antisymmetric tensors of $mathcal{H}^{otimes n}$ and $wedge^{0}mathcal{H} := mathbb{C}$.

Question: Can I use the second approach to define fermionic Fock spaces in an equivalent way, as before? In other words, if I set $mathcal{F}’_{f}(mathcal{H}) := bigoplus_{n=0}^{infty}wedge^{n}mathcal{H}$, does it follow that $mathcal{F}_{f}(mathcal{H}) = mathcal{F}’_{f}(mathcal{H})$? Equivalently: is it possible to prove that every $psi in wedge^{n}mathcal{H}$ can be expressed as $psi = frac{1}{n!}sum_{sigma}epsilon_{sigma}sigma^{*}varphi$ for some $varphi in mathcal{H}^{otimes n}$?

reference request – From Zurab’s integral representation for the Apéry’s constant to almost impossible integrals

I would like to know if the following integrals are known, or in case that aren’t in the literature we can calculate these in closed-form (in terms of elementary and standard functions). I wondered about it when I tried to get variants of an integral representation from a paper of Zurab (identity $(22)$ from (1), that I refer as Zurab’s integral in the title), in next two contexts. If my post is more suitable for Mathematics Stack Exchange or you’ve feedback (in particular if you know these integrals) please add your feedback in comments.

Question 1. The first, just as comparison to (1), calculate if possible $$int_0^{pi}frac{J_0(x)}{binom{1}{x/pi}}dx,tag{1}$$
where $J_0(x)$ denotes the Bessel function. Many thanks.

For Question 1 I tried to combine an integral representation from the Wikipedia Bessel function and calculations from Wolfram Alpha online calculator, in particular I did an attempt to exploit the section Bessel’s integral that refers previous Wikipedia (reference Nico M. Temme, Special Functions: An introduction to the classical functions of mathematical physics (2nd print ed.). New York: Wiley, (1996) pp. 228–231), and the output int x(pi-x)/sin(x) cos(a x) dx that provides me Wolfram Alpha online calculator.

Question 2. I would like to know as reference request or in other case if it is possible to compute the integrals $$c_{n,m}:=int_0^{1/2}frac{x(x-1/2)}{sin^2(2pi x)}, sin(2pi(2m+1)nx)dxtag{2}$$
for integers $mgeq 0$ and integers $ngeq 1$. Many thanks.

Thus for Question 2 if you know a reduction formula or reference for these integrals $(2)$ from the literature helpful here, answer my question as a reference request and I try to search and read from the literature their closed-forms.

If I remember well I could to find just a choice of integration by parts that was suitable in the computations that I did few months ago, but it doesn’t solve the problem, since I didn’t know how to finish my (tedious) calculation to get in closed-form $(2)$. This is just a curiosity that I wondered few months ago, again from Zurab’s integral as starting point in an attempt to combine his integral and a definition of a certain function (and arithmetical functions that I omit here from page 79 of (2)) I wrote $$-frac{7zeta(3)}{8pi^3}=sum_{n=1}^inftysum_{m=0}^inftyfrac{mu(n)chi_1(n)}{n^2}cdotfrac{(-1)^m c_{n,m}}{(2m+1)^2},$$ where $zeta(s)$ is the Riemann zeta function (and those other arithmetic functions from the mentioned article (2)).

References:

(1) Zurab Silagadze, Sums of Generalized Harmonic Series. For Kids from Five to Fifteen, RESONANCE, September 2015.

(2) Manuel Benito, Luis M. Navas and Juan Luis Varona, Möbius inversion from the point of view of arithmetical semigroup flows, Biblioteca de la Revista Matemática Iberoamericana, Proceedings of the “Segundas Jornadas de Teoría de Números” (Madrid, 2007), pp. 63-81.

Asymptotic Time complexity Computation for finding 1’s in binary representation of integer

Below code snippet is used in the book elements of programming to compute the number of 1’s present in the given integer x

    public static short countBits(int x) {
        short numBits = 0;
        while (x != 0) {
            numBits += (x & 1);
            x >>>= 1 ;
        }
        return numBits;
    }

The author explains that time complexity of the above code is O(n) when n is the number of digits present in x in its binary representation.

if we consider the numBits increment statement as a constant time operation and right shift of the given number would be O(n) for each iteration right? since we right shift the x by one digit for every iteration and this operation essentially divides the number(x) by 2 during each iteration

so shouldn’t the time complexity be O(n log n ) since we perform O(n) worst case computations for log(n) times?

Question on the residual representation

Let $P=MN$ be a standard parabolic subgroup $G=SO_n$ and $sigma$ a cuspidal representation of $M$.

Consider the normalized parabolic induced representation $text{Ind}_P^G(sigma|cdot|^z)$ and for sufficiently large $z$, we can define Eisenstein series $E(z,phi)$ for $phi in text{Ind}_P^G(sigma)$. Since $E(z,phi)$ has a meromorphic continuation, let $z_0$ be a simple pole of $E(z,phi)$. Put $mathcal{E}(phi,z_0)$ a residue of $E(z,phi)$ at $z=z_0$.

I am wondering if there are two $phi_1,phi_2 in text{Ind}_P^G(sigma)$ such that $mathcal{E}(phi_1,z_0)=mathcal{E}(phi_2,z_0)$, then $mathcal{E}(phi_1,z)=mathcal{E}(phi_2,z)$ as meromorphic functions on $mathbb{C}$. Is it right?

design – Should an application interface directly with a DSL representation (eg. YAML) vs. interfacing to some intermediate representation?

Consider an application that takes as input some YAML configuration file that defines some network infrastructure and then provisions the appropriate network infrastructure.

Example YAML config:

devices:
  computerA:
    ip: a.b.c.d
    eths:
      - eth1
  computerB:
    ip: a.b.c.d
    eths:
      - eth2
  computerC:
    ip: a.b.c.d
    eths:
      - eth3

network-connections:
   - (computerA.eth1, computerB.eth2)
   - (computerB.eth2, comptuerC.eth3)

What is the best practice for here? Should you application program to a dictionary representation of this YAML file (no objects, simply access data via dictionary) or is it better to create some intermediate representation (objects for “computer”, “connections”, etc..) for the network outlined by this YAML file, and then the applciation would interface to the intermediate represntation? Your thoughts on this would be appreciated.

My inital thoughts:

  1. Programming to the dictionary representation of this YAML would mean that your
    entire application would be littered with access to this dictionary. This
    couples your application to the configuration yaml schema.
  2. If changes to yaml schema are not to be expected, then you can simplify your
    application by not having to create objects. You can avoid several levels of
    indirections (caused by creating objects) => possibly cleaner code base and
    more efficient application?

EDIT: If possible, please provide answer that applies to the more general case than the specific example given above. The example above is only provided to give better idea to the reader about the question I am asking..

Thank you.

javascript – Good representation for list of items that can be grouped together

I am building the frontend of a web app in JavaScript that manages list of past events. It basically has the following requirements:

  • Events are displayed in a vertical list.
  • Events can be moved to arbitrary different positions in the list.
  • Any event can be “connected” to the event before or after it, by clicking an icon that appears between them.
  • When two events are connected, they form a ‘group’.
  • Connecting further events to a group merges the events into the group.
  • Groups move as if they were single items: moving a group moves all the events within that group.
  • Groups can also be disconnected by clicking another icon that appears between events, which will split the group into two groups (or individual events, if only one event in a group was left isolated).

How would you structure the data representation of such a system? i.e. what would the data structure representing the list be, what operations would exist to implement the move and display operations? I’m talking purely on the frontend here.

My naive idea was that the event list is basically an array, where each element of the array is a union type that can be either a group (which is itself an array) or a single item. But that seems inelegant in many ways when it comes to the moving, reading, and split operations. I am wondering if there is a better representation that I have missed that can simplify the implementation of these operations.

Relations between quantum groups at roots of unity, modular representation theory, and physics

I understand that quantum groups at roots of unity are related to physics because they are used in the construction of Reshetikhin-Turaev invariants, conjectured by Witten. Are there other relations of quantum groups at roots of unity to physics? Also, modular representation theory of Lie algebras is related to quantum groups at roots of unity via Andersen-Jantzen-Soergel (Lusztig’s conjecture). Modular representation theory is a very active area of research, and I am wondering if there are relations between results/questions in this area and physics.