Branching Rule for alternating groups

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Let $A_n$ be the alternating group of degree $n$. What is the branching rule for the subgroup $A_n-1subset A_n$, i.e., the structure of the restriction of ordinary irreducible representations of $A_n$ to $A_n-1$? Are there some nice books or references which provide detailed answer to this question?






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    up vote
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    Let $A_n$ be the alternating group of degree $n$. What is the branching rule for the subgroup $A_n-1subset A_n$, i.e., the structure of the restriction of ordinary irreducible representations of $A_n$ to $A_n-1$? Are there some nice books or references which provide detailed answer to this question?






    reference-request gr.group-theory rt.representation-theory







    share|cite|improve this question

























      up vote
      4
      down vote

      favorite









      up vote
      4
      down vote

      favorite











      Let $A_n$ be the alternating group of degree $n$. What is the branching rule for the subgroup $A_n-1subset A_n$, i.e., the structure of the restriction of ordinary irreducible representations of $A_n$ to $A_n-1$? Are there some nice books or references which provide detailed answer to this question?






      reference-request gr.group-theory rt.representation-theory







      share|cite|improve this question















      Let $A_n$ be the alternating group of degree $n$. What is the branching rule for the subgroup $A_n-1subset A_n$, i.e., the structure of the restriction of ordinary irreducible representations of $A_n$ to $A_n-1$? Are there some nice books or references which provide detailed answer to this question?






      reference-request gr.group-theory rt.representation-theory




      reference-request gr.group-theory rt.representation-theory






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      Amritanshu Prasad

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      Xueyi Huang

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          This is answered in Theorem 4 of my paper Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups, with Geetha Thangavelu, which is published in Algebras and Representation Theory, and is also available on the arXiv. See also Ruff, O.: Weight theory for alternating groups. Algebra Colloq. 15(03), 391–404 (2008).



          The alternating groups have two types of representations, those corresponding to mutually conjugate non-self-conjugate pairs $(lambda,lambda')$ which are simply the restriction of the irreducible representation $V_lambda$ or $V_lambda'$ of $S_n$ to $A_n$, and two corresponding to each self-conjugate partition $lambda$, denoted $V_lambda^pm$, which are the two irreducible summands of the irreducible representation $V_lambda$ of $S_n$, when resticted to $A_n$.



          If $lambda$ is a partition of $n$ and $mu$ is a partition of $n-1$, write $muin lambda^-$ if the representation $V_lambda$ of $S_n$ contains the representation $V_mu$ of $S_n-1$ upon restriction, then we have:



          1. If $lambda$ and $mu$ are non-self-conjugate then the representation $V_mu$ of $A_n-1$ is contained in the restriction of $V_lambda$ from $A_n$ to $A_n-1$ if either $muin lambda^-$, or $mu'in lambda^-$.


          2. If $lambda$ is non-self-conjugate and $muin lambda^-$ is self-conjugate, then $V_mu^pm$ are both contained in the restriction of $V_lambda$ from $A_n$ to $A_n-1$.


          3. If $lambda$ is self-conjugate and $muin lambda^-$ is non-self-conjugate, then $V_mu$ is contained in the restriction of both $V_lambda^pm$ from $A_n$ to $A_n-1$.


          4. Finally, if $muin lambda^-$ are both self-conjugate, then $V_mu^+$ is contained in $V_lambda^+$ and $V_mu^-$ is contained in $V_lambda^-$. This result is based on a careful choice of sign in defining the representations $V_lambda^pm$ (deciding which gets the $+$ sign, and which gets the $-$ minus sing among the irreducible $A_n$ representations contained in a self-conjugate representation of $S_n$).


          See the figure below. Please write to me if you would like an e-print of the the published version of the article.



          Bratteli diagram of alternating groups






          share|cite






















          • Thanks for the two references. I have downloaded them. The results are very important and interesting.
            – Xueyi Huang
            11 mins ago










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          1 Answer
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          active

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          active

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          up vote
          3
          down vote



          accepted










          This is answered in Theorem 4 of my paper Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups, with Geetha Thangavelu, which is published in Algebras and Representation Theory, and is also available on the arXiv. See also Ruff, O.: Weight theory for alternating groups. Algebra Colloq. 15(03), 391–404 (2008).



          The alternating groups have two types of representations, those corresponding to mutually conjugate non-self-conjugate pairs $(lambda,lambda')$ which are simply the restriction of the irreducible representation $V_lambda$ or $V_lambda'$ of $S_n$ to $A_n$, and two corresponding to each self-conjugate partition $lambda$, denoted $V_lambda^pm$, which are the two irreducible summands of the irreducible representation $V_lambda$ of $S_n$, when resticted to $A_n$.



          If $lambda$ is a partition of $n$ and $mu$ is a partition of $n-1$, write $muin lambda^-$ if the representation $V_lambda$ of $S_n$ contains the representation $V_mu$ of $S_n-1$ upon restriction, then we have:



          1. If $lambda$ and $mu$ are non-self-conjugate then the representation $V_mu$ of $A_n-1$ is contained in the restriction of $V_lambda$ from $A_n$ to $A_n-1$ if either $muin lambda^-$, or $mu'in lambda^-$.


          2. If $lambda$ is non-self-conjugate and $muin lambda^-$ is self-conjugate, then $V_mu^pm$ are both contained in the restriction of $V_lambda$ from $A_n$ to $A_n-1$.


          3. If $lambda$ is self-conjugate and $muin lambda^-$ is non-self-conjugate, then $V_mu$ is contained in the restriction of both $V_lambda^pm$ from $A_n$ to $A_n-1$.


          4. Finally, if $muin lambda^-$ are both self-conjugate, then $V_mu^+$ is contained in $V_lambda^+$ and $V_mu^-$ is contained in $V_lambda^-$. This result is based on a careful choice of sign in defining the representations $V_lambda^pm$ (deciding which gets the $+$ sign, and which gets the $-$ minus sing among the irreducible $A_n$ representations contained in a self-conjugate representation of $S_n$).


          See the figure below. Please write to me if you would like an e-print of the the published version of the article.



          Bratteli diagram of alternating groups






          share|cite






















          • Thanks for the two references. I have downloaded them. The results are very important and interesting.
            – Xueyi Huang
            11 mins ago














          up vote
          3
          down vote



          accepted










          This is answered in Theorem 4 of my paper Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups, with Geetha Thangavelu, which is published in Algebras and Representation Theory, and is also available on the arXiv. See also Ruff, O.: Weight theory for alternating groups. Algebra Colloq. 15(03), 391–404 (2008).



          The alternating groups have two types of representations, those corresponding to mutually conjugate non-self-conjugate pairs $(lambda,lambda')$ which are simply the restriction of the irreducible representation $V_lambda$ or $V_lambda'$ of $S_n$ to $A_n$, and two corresponding to each self-conjugate partition $lambda$, denoted $V_lambda^pm$, which are the two irreducible summands of the irreducible representation $V_lambda$ of $S_n$, when resticted to $A_n$.



          If $lambda$ is a partition of $n$ and $mu$ is a partition of $n-1$, write $muin lambda^-$ if the representation $V_lambda$ of $S_n$ contains the representation $V_mu$ of $S_n-1$ upon restriction, then we have:



          1. If $lambda$ and $mu$ are non-self-conjugate then the representation $V_mu$ of $A_n-1$ is contained in the restriction of $V_lambda$ from $A_n$ to $A_n-1$ if either $muin lambda^-$, or $mu'in lambda^-$.


          2. If $lambda$ is non-self-conjugate and $muin lambda^-$ is self-conjugate, then $V_mu^pm$ are both contained in the restriction of $V_lambda$ from $A_n$ to $A_n-1$.


          3. If $lambda$ is self-conjugate and $muin lambda^-$ is non-self-conjugate, then $V_mu$ is contained in the restriction of both $V_lambda^pm$ from $A_n$ to $A_n-1$.


          4. Finally, if $muin lambda^-$ are both self-conjugate, then $V_mu^+$ is contained in $V_lambda^+$ and $V_mu^-$ is contained in $V_lambda^-$. This result is based on a careful choice of sign in defining the representations $V_lambda^pm$ (deciding which gets the $+$ sign, and which gets the $-$ minus sing among the irreducible $A_n$ representations contained in a self-conjugate representation of $S_n$).


          See the figure below. Please write to me if you would like an e-print of the the published version of the article.



          Bratteli diagram of alternating groups






          share|cite






















          • Thanks for the two references. I have downloaded them. The results are very important and interesting.
            – Xueyi Huang
            11 mins ago












          up vote
          3
          down vote



          accepted







          up vote
          3
          down vote



          accepted






          This is answered in Theorem 4 of my paper Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups, with Geetha Thangavelu, which is published in Algebras and Representation Theory, and is also available on the arXiv. See also Ruff, O.: Weight theory for alternating groups. Algebra Colloq. 15(03), 391–404 (2008).



          The alternating groups have two types of representations, those corresponding to mutually conjugate non-self-conjugate pairs $(lambda,lambda')$ which are simply the restriction of the irreducible representation $V_lambda$ or $V_lambda'$ of $S_n$ to $A_n$, and two corresponding to each self-conjugate partition $lambda$, denoted $V_lambda^pm$, which are the two irreducible summands of the irreducible representation $V_lambda$ of $S_n$, when resticted to $A_n$.



          If $lambda$ is a partition of $n$ and $mu$ is a partition of $n-1$, write $muin lambda^-$ if the representation $V_lambda$ of $S_n$ contains the representation $V_mu$ of $S_n-1$ upon restriction, then we have:



          1. If $lambda$ and $mu$ are non-self-conjugate then the representation $V_mu$ of $A_n-1$ is contained in the restriction of $V_lambda$ from $A_n$ to $A_n-1$ if either $muin lambda^-$, or $mu'in lambda^-$.


          2. If $lambda$ is non-self-conjugate and $muin lambda^-$ is self-conjugate, then $V_mu^pm$ are both contained in the restriction of $V_lambda$ from $A_n$ to $A_n-1$.


          3. If $lambda$ is self-conjugate and $muin lambda^-$ is non-self-conjugate, then $V_mu$ is contained in the restriction of both $V_lambda^pm$ from $A_n$ to $A_n-1$.


          4. Finally, if $muin lambda^-$ are both self-conjugate, then $V_mu^+$ is contained in $V_lambda^+$ and $V_mu^-$ is contained in $V_lambda^-$. This result is based on a careful choice of sign in defining the representations $V_lambda^pm$ (deciding which gets the $+$ sign, and which gets the $-$ minus sing among the irreducible $A_n$ representations contained in a self-conjugate representation of $S_n$).


          See the figure below. Please write to me if you would like an e-print of the the published version of the article.



          Bratteli diagram of alternating groups






          share|cite














          This is answered in Theorem 4 of my paper Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups, with Geetha Thangavelu, which is published in Algebras and Representation Theory, and is also available on the arXiv. See also Ruff, O.: Weight theory for alternating groups. Algebra Colloq. 15(03), 391–404 (2008).



          The alternating groups have two types of representations, those corresponding to mutually conjugate non-self-conjugate pairs $(lambda,lambda')$ which are simply the restriction of the irreducible representation $V_lambda$ or $V_lambda'$ of $S_n$ to $A_n$, and two corresponding to each self-conjugate partition $lambda$, denoted $V_lambda^pm$, which are the two irreducible summands of the irreducible representation $V_lambda$ of $S_n$, when resticted to $A_n$.



          If $lambda$ is a partition of $n$ and $mu$ is a partition of $n-1$, write $muin lambda^-$ if the representation $V_lambda$ of $S_n$ contains the representation $V_mu$ of $S_n-1$ upon restriction, then we have:



          1. If $lambda$ and $mu$ are non-self-conjugate then the representation $V_mu$ of $A_n-1$ is contained in the restriction of $V_lambda$ from $A_n$ to $A_n-1$ if either $muin lambda^-$, or $mu'in lambda^-$.


          2. If $lambda$ is non-self-conjugate and $muin lambda^-$ is self-conjugate, then $V_mu^pm$ are both contained in the restriction of $V_lambda$ from $A_n$ to $A_n-1$.


          3. If $lambda$ is self-conjugate and $muin lambda^-$ is non-self-conjugate, then $V_mu$ is contained in the restriction of both $V_lambda^pm$ from $A_n$ to $A_n-1$.


          4. Finally, if $muin lambda^-$ are both self-conjugate, then $V_mu^+$ is contained in $V_lambda^+$ and $V_mu^-$ is contained in $V_lambda^-$. This result is based on a careful choice of sign in defining the representations $V_lambda^pm$ (deciding which gets the $+$ sign, and which gets the $-$ minus sing among the irreducible $A_n$ representations contained in a self-conjugate representation of $S_n$).


          See the figure below. Please write to me if you would like an e-print of the the published version of the article.



          Bratteli diagram of alternating groups







          share|cite














          share|cite



          share|cite








          edited 2 hours ago

























          answered 2 hours ago









          Amritanshu Prasad

          3,7892437




          3,7892437











          • Thanks for the two references. I have downloaded them. The results are very important and interesting.
            – Xueyi Huang
            11 mins ago
















          • Thanks for the two references. I have downloaded them. The results are very important and interesting.
            – Xueyi Huang
            11 mins ago















          Thanks for the two references. I have downloaded them. The results are very important and interesting.
          – Xueyi Huang
          11 mins ago




          Thanks for the two references. I have downloaded them. The results are very important and interesting.
          – Xueyi Huang
          11 mins ago

















           

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