How many categories does this point representation represent?

I came across these questions and I do not know how to answer them. My guess is 3 categories.

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representation theory – Invariants in the symmetric algebra of a module

Let $ mathfrak {g} $ to be a complex semi-simple Lie algebra of finite dimension, and $ V $ a finite dimension $ mathfrak {g} $-module. then $ mathfrak {g} $ also acts on symmetric algebra $ S (V) $.

Is there a description of invariants $ S (V) ^ mathfrak {g} $?

Yes $ V $ is the standard module of classical algebras, so it is reduced to the fundamental theorems of invariant theory. Is there anything in literature of this kind more general? Is it known, at least for $ mathfrak {sl} _2 $?

Theory of Representation – Semi-Simple Lie Algebra Modules with Dimensional Weight Spaces at $ 1

Given a semi-simple complex Lie algebra $ frak {g} $ of rank $ r $, with Chevally generators $ E_i, F_i, K_i $. Let $ V $ to be a finite dimensional representation of $ frak {g} $ such as each weight space of $ V $ is $ 1 $-dimensional. Let $ (i_1, dots, i_k) $ to be an ordered set of elements of $ {1, dots, r } $ (allowing rehearsals), and let $ {j_1, points, j_k } $ to be a permutation of $ {1, dots, r } $. For $ v $ a higher weight of $ v $, the elements
$$
F_ {i_1} circ F_ {i_2} cdots circ F_ {i_k} (v), ~~ text {and} ~~~ circ F_ {j_1} circ F_ {j_2} cdots F_ {j_k} (v)
$$

must have the same weight. Thus, by our hypothesis, they must differ by a scalar multiple. Will this scalar multiple always be an integer?

theory of representation – Extension of irreducible tempered representations of $ GL_n ( mathbb Q_p) $

Let $ V_1, V_2 $ be two irreducible temperate representations permissible to $ GL_n ( mathbb Q_p) $, how to calculate all the extensions $ V $ (as smooth representations of $ GL_n ( mathbb Q_p) $) of $ V_1 $ by $ V_2 $ ?

Must such a representation $ V $ to be eligible and temperate?

Number Theory – Decrease of Matrix Coefficients of Non-Quenched Representation

A Cowling – Haagerup – Howe theorem gives an effective decay rate of matrix coefficients of a tempered representation. $ pi $ of a semi-simple algebraic $ G $ in terms of Harish-Chandra $ Xi $ function, form
$$ langle pi (g) u, v rangle ll_ {u, v} Xi (g), quad u, v in pi setminus pi ^ G, g in G. $$
Is there a similar terminal, possibly lower, in the case of non-tempered representations (possibly of the form above but $ Xi (g) $ is replaced by $ Xi (g) ^ {1- delta} $ for some people $ 0 < delta <1 $)?

I am particularly interested in $ mathrm {GL} (n) $ and $ pi $ to be an irreducible automorphic representation of it. For $ n = $ 2 we know that such $ delta $ exists (spectral gap) due to the works of Selberg, Gelbart – Jacquet, Kim – Shahidi.

Thanks in advance!

graph theory – Iterated inverted structures: polynomial representation of integer partitioning of pre-images in Sigma matrices (reference request)

I study structures of pre-image functions iterated over a finite set.

The main structure that interests me, the Sigma matrix, is derived from a matrix enumerating sets of pre-images per element at an increasing inverse depth. This intermediate matrix is ​​the "preimage matrix" (represented by $ P $ below)

$ P = left ( begin {array} {cccc}
f ^ {- 1} (x_ {1}) & f ^ {- 2} (x_ {1}) & cdots & f ^ {- n} (x_ {1}) \
f ^ {- 1} (x_ {2}) & ddots \
vdots \
f ^ {- 1} (x_ {n}) & & f ^ {- n} (x_ {n})
end {array} right) $

We then look at the sizes such matrix elements.
Give the matrix "sigma" $ Sigma $ with entries

$ Sigma = left ( begin {array} {cccc}
mid f ^ {- 1} (x_ {1}) mid & cdots & mid f ^ {- n} (x_ {1}) mid \
& ddots \
vdots \
mid f ^ {- 1} (x_ {n}) mid & & mid f ^ {- n} (x_ {n}) mid
end {array} right) $

  • Result 1Each column forms an entire partition of n = dom (f) since each column of the sigma matrix has the sum n.

Leave each column of $ Sigma $ be represented by his own polynomial where: if $ col (1) = (a, b, c) $ then the associated polynomial for $ Sigma_ {X1} $ is
$ y_ {1} = ax ^ {0} + bx ^ {1} + cx ^ {2} $

  • Result 2: Now taking all the columns, for any size sigma matrix (on any size domain) and deriving all the polynomials, they have a solution to $ (1, n) $.

Question: The seems always be one and only one solution to these systems of equations. I would appreciate a reference to this type of exploration

For more information, please refer to my last question about sigma matrices: previous question

representation theory – strange formulas that gave rise to the Koszul duality

According to p.8 of the note KOSZUL DUALITY AND APPLICATIONS IN
THEORY OF REPRESENTATION
by Geordie Williamson.
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Let $ M ( eta) $ to be the Verma weight module $ eta $, $ L ( eta) $ to be his single simple quotient and $ w_0 $ to be the longest element $ W $.

The strange formula in our notation is
$ (M (x cdot 0): L (y cdot 0)) = sum_ {i ge 0} mathrm {Ext} _ { mathcal {O}} ^ i (M (w_0x cdot 0) : L (w_0y cdot 0)) $ for $ x, y in W $.

Let $ mu $ to be an integral weight, anti-dominant, $ Delta $ be the whole of the simple roots, $ Sigma = { alpha in Delta: langle mu + rho, alpha ^ lor row = 0 } $ and $ W ^ { Sigma} = {w in W: w <ws_ alpha, forall alpha in Sigma } $.

Do we still have $ (M (x cdot mu): L (y cdot mu)) = sum_ {i ge 0} mathrm {Ext} _ { mathcal {O}} ^ i (M (w_0x cdot mu): L (w_0y cdot mu)) $ for $ x, y in W ^ Sigma $?

Theory of representation – $ mathfrak {g} $ hardness – module $ mathfrak {o} (k) / ad ( mathfrak {g}) $

For a simple lie algebra $ mathfrak {g} $, to define $ mathfrak {o} (k) $ to be the orthogonal lie algebra with respect to the Killing form.

In the proof of Theorem 2 of the following article, https://arxiv.org/pdf/math/0407240.pdf, the author mentions that the following is true,

the $ mathfrak {g} $ module $ mathfrak {o} (k) / ad ( mathfrak {g}) $ is irreducible if $ mathfrak {g} $ is not type A whereas it is a direct sum $ W oplus W ^ * $for some non-auto-dual modules $ W $ if $ mathfrak {g} $ is type A.

I can not see why this is true. Can any one provide proof or indicate references?

I have also requested it here – https://math.stackexchange.com/questions/3335347/irreducibility-of-the-mathfrakg-module-mathfrakok-ad-mathfrakg

Theory of Representation – Kazhdan-Lusztig Parabolic Conjecture

Humphreys provides a reference about the Kazhdan-Lusztig conjecture for arbitrary weights in $ mathfrak {h} ^ * $which is under the name: Characteristics of irreducible modules with
highest non-critical weights relative to affine Lie algebras

In other words, for complex semi-simple Lie algebra, the problem of expression $ mathrm {ch} L ( lambda) $ in terms of $ mathrm {ch} M ( mu) $ is completely answered, where $ M ( eta) $ is the Verma module with weight $ eta $ and $ L ( eta) $ is his single simple quotient.

What about the problem to express $ mathrm {ch} L ( lambda) $ in terms of $ mathrm {ch} M_I ( mu) $, or $ M_I ( eta) $ is the parabolic module of Verma with weights $ eta $ and $ L ( eta) $ is his single simple quotient. Is it always a guess or is it simply derived from the expression of $ mathrm {ch} L ( lambda) $ in terms of $ mathrm {ch} M ( mu) $?

Number Theory – Representation of natural numbers as sums of powers of distinct numbers

Find the smallest number $ n $ so that almost all natural numbers can be represented by the sum $$ a_1 ^ {a_ {p (1)}} + a_2 ^ {a_ {p (2)}} + points + a_n ^ {a_ {p (n)}} $$or $ a_1, dots, a_n $ are distinct natural numbers in pairs and $ p $ is a permutation of the whole $ {1, dots, n } $.


The problem was raised on 24.03.2019 by Jacek Jurewicz on page 95 of volume 2 of the Scottish Lviv book.

The price: A personal congratulation πŸ™‚