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        Questions tagged [rt.representation-theory]

        Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

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        4
        votes
        1answer
        57 views

        Integral of Schur functions over $SU(N)$ instead of $U(N)$

        Schur functions are irreducible characters of the unitary group $U(N)$. This implies $$ \int_{{U}(N)}s_\lambda(u)\overline{s_\mu(u)}du=\delta_{\lambda\mu},$$ where the overline means complex ...
        2
        votes
        0answers
        32 views

        Determining the groups compatible with given fusion rules

        Say I have an unknown group $G$ with a simple, real, faithful representation $\mathbf n$ such that $$ {\mathbf n}\otimes{\mathbf n} \approx {\mathbf 1}_s\oplus{\mathbf a}_s\oplus{\mathbf b}_s\oplus{\...
        3
        votes
        1answer
        113 views

        Complete reducibility and field extension

        Let $\pi$ be a representation of a Lie algebra $L$ in a finite-dimensional linear space $V$ over the field $F$. Let $K$ be a field extension of $F$. Let $\pi_K=\pi\otimes K$ be the corresponding ...
        0
        votes
        0answers
        60 views

        Root system for semisimple Lie algebras [on hold]

        Let $L$ be a semisimple Lie algebra. We can have a decomposition $L=\oplus_{\alpha=1}^tL_\alpha$ where each $L_\alpha$ is a simple ideal. It is well known that if we fix a maximal toral algebra $H$ of ...
        2
        votes
        2answers
        102 views

        Greatest element of ${}^IW$

        Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Let $W$ be the associated Weyl group and let $\Phi$ be its root system. We write $\Phi^+...
        1
        vote
        0answers
        43 views

        $\mathrm{Ext}^1$-ordering on ${}^IW^{\Sigma_\mu}$

        Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Let $W$ be the associated Weyl group and let $\Phi$ be its root system. We write $\Phi^+...
        0
        votes
        0answers
        65 views

        A characterization of reflections on Euclidean space [migrated]

        In the book "Introduction to Lie algebras and Representation theory" by J. Humphreys page 42, the author proves the following lemma: Let $E$ be a finite dimensional Euclidean space over $\mathbb R$ ...
        3
        votes
        1answer
        108 views

        Decomposing tensor powers of the fundamental representation of exceptional Lie algebras

        For the $A$-series, tensor powers of the fundamental representation of $\frak{sl}_n$ decompose into irreducibles according to a certain Young diagram/ partition formula. This inspires, for example, ...
        4
        votes
        1answer
        186 views

        Verma module and vanishing of extension groups

        Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Let $W$ be the associated Weyl group and let $\Phi$ be its root system. We write $\Phi^+...
        3
        votes
        0answers
        49 views

        Representation theoretic characterisation of symmetric spaces

        Let $G$ be a simple compact Lie group and $H$ a closed subgroup. Let $\mathfrak{h}\subset \mathfrak{g}$ denote the corresponding Lie algebras. Let $\mathfrak{m}$ be an orthogonal complement to $\...
        10
        votes
        1answer
        147 views

        What subalgebras of $\mathfrak{so}(2n)$ or $\mathfrak{sp}(2n)$ are orthogonal to the centre of $\mathfrak{gl}(n)$?

        Consider the Lie algebra inclusions $\mathfrak{gl}(n) \subset \mathfrak{so}(2n)$ and $\mathfrak{gl}(n) \subset \mathfrak{sp}(2n)$. Let $\mathfrak{c} \subset \mathfrak{gl}(n)$ denote the centre. ...
        4
        votes
        0answers
        73 views

        Branching rules for E6 into SU(3)^3

        I am very confused about what are the branching rules for representations of $E6$ into a $SU(3)\times SU(3)\times SU(3)$ subgroup. At least in the physics literature, there seems to be a serious ...
        5
        votes
        2answers
        142 views

        Irreducible factors of primitive permutation group representation

        Consider a primitive permutation group $\Gamma\subseteq\mathrm{Sym}(N)$ on the $n$-element set $N=\{1,...,n\}$, that is, $\Gamma$ does not preserve any non-trivial partition of $N$. Consider the ...
        4
        votes
        0answers
        191 views

        Computing the $2$-adic volume of a special orthogonal group

        Let $n \geq 0$ be an integer, let $A = (a_{ij})$ be the $(2n+1) \times (2n+1)$ matrix defined by $a_{ij} = 0$ unless $i + j = 2n+2$, in which case $a_{ij} = 1$. Let $G$ be the group scheme over $\...
        2
        votes
        0answers
        56 views

        $End_A(M)$ for representation-finite $A$ and indecomposable $M$

        Given a representation-finite finite dimensional algebra $A$ and an indecomposable module $M$. Question: What possible algebras can occur as $End_A(M)$? Is it always tame? Note than when $End_A(M)$...

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