All Questions
Tagged with modules rt.representation-theory
121 questions
0
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Endomorphism algebra of equivariant maps of isotypic module
Let $A$ be a simple Artinian $K$-algebra with a minimal left ideal $M$. Here, $M$ can be viewed as a simple left $A$ module, and, by Schur's lemma, $D=\text{End}_A(M)$ is a $K$-algebra. By Wedderburn-...
- 143
3
votes
0
answers
107
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Dimension of hom spaces between indecomposable modules
Undergraduate-Level Background
Let $A$ be an Artin algebra over an algebraically closed field $k$, and let $C = Rep(A)$ denotes the category of $k$-linear, $k$-finite dimensional representations of $A$...
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1
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301
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If a bimodule is "generated" by single elements, must the elements be conjugate?
Let $A$ and $B$ be Artin $k$-algebras for a commutative artinian ring $k$ (e.g. $A$ and $B$ are finite dimensional $k$-algebras for a field $k$). Let $M$ be an $A$-$B$-bimodule of finite length over $...
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1
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339
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If the Hom-space of finite length modules is generated by single elements, must the elements be conjugate?
Let $A$ be an Artin $k$-algebra for a commutative artinian ring $k$ (e.g. $A$ is a finite dimensional algebra over a field $k$). Let $X,Y$ be finite length left $A$-modules. If $\text{Hom}_A(X,Y)$ is ...
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288
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Representation functor on modules
Let $k$ be a field and $A$ a unital associative $k$-algebra.
The representation functor associates, to each object in non-commutative geometry, a genuine geometric object on the representation variety ...
- 985
3
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0
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107
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Auslander-Reiten sequences where irreducible morphisms are all epi/mono
Let's work in the setting of modules over an Artin algebra $A$, or a finite-dimensional $k$-algebra $A$, or if you like, modules over a connected quiver $Q$ without oriented cycles.
Let $M$ be such a ...
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2
votes
2
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139
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Infinite radical ideal cubed equals zero for tame hereditary Artin algebras
Let $A$ be a tame hereditary Artin algbera and mod$A$ the category of finitely generated (left) $A$-modules. Further, let rad$_A$ be the radical ideal of mod$A$, which is the smallest ideal containing ...
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4
votes
1
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198
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Does hereditary and connected imply that the underlying ring $k$ of a $k$-algebra is a field?
All rings are assumed to be associative and have a 1. Let $k$ be a commutative artininan ring and $R$ a finitely generated $k$-algebra. Is it true that if $R$ is connected and hereditary, then $k$ is ...
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4
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1
answer
160
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Symmetric tensor of highest weight modules for $\mathrm{SU}(2)$
Let $V_i$ be the $(i+1)$-dimensional representation of the special unitary group $\mathrm{SU}(2)$ with the highest weight $i$. Is there any uniform way to compute the irreducible decomposition for the ...
- 951
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1
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264
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Schur functors = Weyl functors in characteristic zero?
I asked this question on Math Stack Exchange https://math.stackexchange.com/questions/4789924/schur-functors-weyl-functors-in-characteristic-zero, but I got no answers, so I ask the same question here....
- 127
2
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100
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Pairs of ideals in an abelian category similar to torsion pairs
Let $\mathcal{A}$ be an abelian category. In the context of my work I am considering pairs of ideals $(\mathcal{I}, \mathcal{J})$ in $\mathcal{A}$ with the following properties:
$\quad \mathcal{I} \...
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11
votes
1
answer
159
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Are all indecomposable $\mathbb{Z}_+$-modules over the character ring of a group, character rings of a subgroup?
A $\mathbb Z_+$-algebra is an algebra $A$ over $\mathbb C$ with given basis $\{v_i\}$ such that
$$v_iv_j=\sum_k n_{ijk}v_k,\hspace{10mm}n_{ijk}\in\mathbb Z_{\geq0}.$$
An example of such an object is ...
- 301
3
votes
1
answer
355
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A generalisation of induced representations
Let $G$ be a finite group, and $H\subseteq G$ a subgroup. Let $F$ be a field. Let $W$ be a finite-dimensional $F[H]$-module. Let $T$ be a left transversal of $H$ in $G$. Then we can define:
$W^G=\sum_{...
- 1,433
2
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79
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Almost split sequences for symmetric algebras
Let $k$ be an algebraically closed field and $A$ be a symmetric algebra.
I want to know how to compute almost split sequences ending at a non-projective indecomposable right $A$-module $X$.
Question: ...
- 21
2
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1
answer
105
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Baur-Monk quantifier elimination (BG-invariants in 1-free variable)
$\DeclareMathOperator\Inv{Inv}$Baur-Monk quantifier elimination implies that a sentence in the language of modules is a combination of BG invariant statements.
A BG invariant sentence is a boolean ...
10
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1
answer
1k
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Clebsch–Gordan decomposition formula for algebraic groups
$\DeclareMathOperator\SL{SL}$There is a well-known Clebsch–Gordan decomposition formula for irreducible representations of $\SL_2$. If $V_n$ denotes the unique $n+1$-dimensional irreducible ...
- 370
4
votes
1
answer
107
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For a finite dimensional $k$-Algebra $A$ does infinite representation type imply $(\text{rad}_A^{\omega})^2 \neq 0$?
Let $k$ be a field, $A$ a finite dimensional $k$-Algebra, $\text{mod}\,A$ the category of finite dimensional left $A$-modules and $\text{rad}_A$ the collection of radical morphisms in $\text{mod}\,A$. ...
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3
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99
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Decidability of theory of modules over a ring of finite representation type
I have read from Mike Prest's model theory for modules (London lecture note series) chapter 17 that a Ring of finite representation type has a decidable theory of modules. Here decidability was ...
- 31
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45
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Extending $G$-closed sets to permutation bases of a permutation $RG$-module
I'm curious if there are any papers or results about the following scenario:
Let $R$ be a commutative ring (I'm interested in particular in the $R = \mathbb{Z}$ case, but fields are okay too), $G$ a ...
- 175
7
votes
1
answer
284
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Does the choice of the algebraically closed field of characteristic $p$ have influence on the module category?
Let $G$ be a finite group and $p$ be a prime number dividing $|G|$.
Let $k$ be the algebraic closure of $\mathbb{F}_p$.
Let $K$ be another algebraically closed field of characteristic $p$ which is not ...
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5
votes
1
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244
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What are the indecomposable modules over $\mathbb{F}_2(C_2\times C_2)$?
Let $C_2$ be the cyclic group of order $2$ and $\mathbb{F}_2$ the field with $2$ elements. Consider the group algebra $A:= \mathbb{F}_2 (C_2\times C_2)$. It is well-known that $A$ has infinite ...
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3
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1
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171
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Modules with special properties
$\DeclareMathOperator\End{End}$Let $A$ be a finite dimensional algebra and $M$ an indecomposable (right) module with the property that every nilpotent element of $\End_A(M)$ annihilates the socle $\...
- 26.5k
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1
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187
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Semi simplicity over commutative algebras over non-algebraically closed fields
I have already posted this on stackexchange
I have a question:
If k is an arbitrary field then is it true that if $M$ a finite dimensional $k[x, y]$ is semisimple as a $k[x]$ module and also as a $...
- 159
5
votes
1
answer
707
views
Absolutely irreducible representation and splitting field
Let $A$ be a finite-dimensional algebra over a field $F$. A representation $M$ of $A$ is called absolutely irreducible if $M\otimes_FE$ is irreducible as a representation of $A\otimes_FE$ for all ...
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votes
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275
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Split epimorphism of modules - does the finite case imply the infinite case?
Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Suppose further we have an ...
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65
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Is it possible to describe a $k$-basis for $M\otimes_{kH}N$ when $M$ is a $k[G\times H]$-module and $N$ is a $k[H\times K]$-module?
Suppose $k$ is a field for Let $M$ be a finitely-generated a $k[G\times H]$-module and let $N$ be a finitely-generated $k[H\times K]$-module. Then in particular, $M$ and $N$ are finite-dimensional $k$-...
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1
answer
240
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Split monomorphisms of modules - does the finite case imply the infinite case?
Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Further $X\subseteq M$ and for ...
- 696
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221
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Group cohomology of $\mathbb{Z}$ vs $\mathbb{Z}_p$
Let $M$ be a continuous representation of $\mathbb{Z}_p$ over $\mathbb{F}_p$, likely infinite-dimensional.
There is the inflation map of group cohomology $H^*_{\text{cts}}(\mathbb{Z}_p, M) \rightarrow ...
- 689
3
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1
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181
views
Is this concept of a left-abelian category studied?
A category is abelian if it is preadditive and
it has a zero object,
it has all binary biproducts,
it has all kernels and cokernels, and
all monomorphisms and epimorphisms are normal.
Now we ...
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5
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1
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234
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Concept of an exact ideal of a module category
Let $R$ be a ring and $\text{Mod}\,R$ the category of (left) $R$-modules. Consider an ideal $\mathcal{I}$ of $\text{Mod}\,R$. For $R$-modules $X$ and $Y$ let $\mathcal{I}(X,Y)$ be the collection of ...
- 696
3
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1
answer
182
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On the definition and an example of silting/tilting subcategories in a triangulated categories according to a paper by Aihara and Iyama
In the paper "Silting mutation in triangulated categories" by Aihara and Iyama, I stumbled upon this nice definition( Definition 2.1) of a tilting/silting subcategory of a triangulated ...
- 83
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0
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93
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Is there a category of "chains of modules" that behaves well with taking direct limits?
I came up with the following definition of a category of certain "chains of modules" and want to know if this concept is already known and studied.
Let $R$ be ring. An object in our category ...
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4
votes
0
answers
187
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Generalization of the second Brauer-Thrall conjecture for arbitrary Artin algebras
Let $k$ be a field with infinite cardinality and $A$ a finite dimensional $k$-Algebra. The second Brauer-Thrall conjecture states the following: There are infinitely many natural numbers $n_1<n_2&...
- 696
5
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0
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241
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Finite-dimensional irreducible representations of 2-loop quiver
What are the finite-dimensional irreducible representations of the quiver with one vertex and two loops?
- 491
1
vote
1
answer
395
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Making use of extra symmetries; more examples?
TL; DR.
In representation theory, it's nice to decompose a given representation into smaller ones. One technique is by utilizing extra symmetries. Explicit examples come from compact groups, and I ...
- 5,230
2
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0
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116
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Loewy structure of $S_4$
How to deduce the Loewy Structure of $kS_4$ where $k$ has characteristic 2. I can compute the Cartan matrix and Decomposition matrix with Brauer Characters without difficulties. But when it comes to ...
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1
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270
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Thin representations for quiver algebras
A representation $M$ of a quiver is called thin when $M$ has a dimension vector consisting only of 0 or 1 entries.
When $A=kQ$ is a path algebra for a tree $Q$, then there is the nice result that ...
- 26.5k
2
votes
1
answer
197
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Top and bottom composition factors of $M$ are isomorphic
Let $k$ be a field and $N$ a finite group. Let $M$ be a projective indecomposable $kN$-module. Since the algebra $kN$ is symmetric, it follows that the top and bottom composition factors of $M$ are ...
- 51
5
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1
answer
211
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Simple quotients of a triple tensor product
Let $\mathcal{H}$ be a Hopf algebra over $\mathbb{C}$. Let also $V_1, V_2, V_3$ finite-dimensional simple modules over $\mathcal{H}$ and $Q$ be a simple quotient of $V_1\otimes V_2\otimes V_3$. Is it ...
- 353
2
votes
1
answer
160
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MAGMA-question concerning the transformation of a $kG$ -module $M$ into a right ideal of the group algebra
Let $G$ be a finite group and $k$ be a finite field of characteristic $p>0$ such that $p\mid |G|$.
Let $M$ be a $kG$-module which has an embedding $M\hookrightarrow kG^{reg}$ into the regular $kG$-...
- 1,755
6
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0
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58
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Definition of modular Howe correspondence
Let $(G,G')$ be a pair of mutually centralized subgroups of a symplectic group $Sp_n(\mathbb{F}_q)$ (called a dual pair), and let $\omega_{G,G'}$ be the restriction of the Weil representation (with ...
- 93
4
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64
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Why does this cluster tilting object form a local slice?
I am reading the paper "Cluster automrphisms", here is the link: http://prospero.dmat.usherbrooke.ca/ibrahim/publications/Cluster_Automorphisms.pdf
In the proof of lemma 3.1 I am stuck: For ...
- 775
1
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0
answers
46
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What is the maximal weight submodule of $\text{Hom}_{\mathfrak{g}}(M,N)$?
Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$. Fix a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. For ...
- 507
3
votes
1
answer
256
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Mackey theory for semidirect products: equivalence between constructions for modules
I am learning about the so-called "Mackey Machine" for unitary irreps of semidirect products of locally compact groups. Let $G = N \rtimes H$ where $N$ is a closed normal abelian subgroup and $H$ is a ...
- 399
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1
answer
52
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Does H-supplmented module have D2?
A module $M$ is called H-supplemented if for every submodule $N$ of $M$ there exists a direct summand $D$ of $M$ such that $M = N + X$ if and only if $M = D + X$ for every submodule $X$ of $M$.
A ...
4
votes
1
answer
159
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Effect of extending scalars on maps of modules
Let $k$ be a field and $R$ be a $k$-algebra. Let $M$ and $N$ be left $R$-modules. Finally, let $\ell$ be a field extension of $k$. We thus have an $\ell$-algebra $\ell \otimes R$, and both $\ell \...
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4
votes
1
answer
135
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Global splitting field for algebras
Let $A$ be a finite dimensional algebra.
A field $K$ is a splitting field for an indecomposable $A$-module $M$ in case the local algebra $End_A(M)/(rad(End_A(M))$ is 1-dimensional.
$K$ is called a ...
- 26.5k
3
votes
1
answer
244
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Left module which cannot be made into a bimodule?
Let $A$ be a noncommutative unital algebra, defined over $\mathbb{C}$ say. What is an example of a left $A$-module $M$ that does not admit a right $A$-module structure giving $M$ the structure of a ...
6
votes
2
answers
487
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Adjoints for radical and socle functors
Let $R$ be a ring and $M$ be a $R$-module. Let $rad(M)$ be the radical of $M$, that is, the intersection of all maximal submodules of $M$. Moreover, let $soc(M)$ be the socle of $M$, that is, the sum ...
- 353
4
votes
1
answer
618
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Parabolic Kazhdan-Lusztig Conjecture
In this post, Humphreys provides a reference concerning about Kazhdan-Lusztig Conjecture for arbitrary weights in $\mathfrak{h}^*$, which is under the name: Kashiwara and Tanisaki - Characters of ...
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