direct sum of representation of product groups












1












$begingroup$


Given two finite groups $G_1$ and $G_2$, and some representations $rho_1: G_1 to V_1$ and $rho_2: G_2 to V_2$, it seems the standard way to create a representation for $G_1 times G_2$ is to use the tensor product
$$rho_1(g_1) otimes rho_2(g_2) quad g_1,g_2 in G_1,G_2.$$
It seems to me that one could also use the direct sum
$rho_1(g_1) oplus rho_2(g_2)$,
because the blocks in the matrix form of the representation do not interact and one gets the desired effect. Given that this representation could have a lower dimension than using tensor product, why is it not used?










share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    Given two finite groups $G_1$ and $G_2$, and some representations $rho_1: G_1 to V_1$ and $rho_2: G_2 to V_2$, it seems the standard way to create a representation for $G_1 times G_2$ is to use the tensor product
    $$rho_1(g_1) otimes rho_2(g_2) quad g_1,g_2 in G_1,G_2.$$
    It seems to me that one could also use the direct sum
    $rho_1(g_1) oplus rho_2(g_2)$,
    because the blocks in the matrix form of the representation do not interact and one gets the desired effect. Given that this representation could have a lower dimension than using tensor product, why is it not used?










    share|cite|improve this question









    $endgroup$















      1












      1








      1


      1



      $begingroup$


      Given two finite groups $G_1$ and $G_2$, and some representations $rho_1: G_1 to V_1$ and $rho_2: G_2 to V_2$, it seems the standard way to create a representation for $G_1 times G_2$ is to use the tensor product
      $$rho_1(g_1) otimes rho_2(g_2) quad g_1,g_2 in G_1,G_2.$$
      It seems to me that one could also use the direct sum
      $rho_1(g_1) oplus rho_2(g_2)$,
      because the blocks in the matrix form of the representation do not interact and one gets the desired effect. Given that this representation could have a lower dimension than using tensor product, why is it not used?










      share|cite|improve this question









      $endgroup$




      Given two finite groups $G_1$ and $G_2$, and some representations $rho_1: G_1 to V_1$ and $rho_2: G_2 to V_2$, it seems the standard way to create a representation for $G_1 times G_2$ is to use the tensor product
      $$rho_1(g_1) otimes rho_2(g_2) quad g_1,g_2 in G_1,G_2.$$
      It seems to me that one could also use the direct sum
      $rho_1(g_1) oplus rho_2(g_2)$,
      because the blocks in the matrix form of the representation do not interact and one gets the desired effect. Given that this representation could have a lower dimension than using tensor product, why is it not used?







      group-theory finite-groups representation-theory






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 4 hours ago









      self-educatorself-educator

      4611




      4611






















          1 Answer
          1






          active

          oldest

          votes


















          6












          $begingroup$

          When $V_1$ and $V_2$ are representations of $G_1$ and $G_2$ respectively, I'll use $V_1 boxtimes V_2$ to mean the representation of $G_1 times G_2$ with underlying vector space $V_1 otimes_{mathbb{C}} V_2$, and $V_1 boxplus V_2$ to mean the representation of $G_1 times G_2$ with underlying vector space $V_1 oplus V_2$.



          If $V$ is an irreducible representation of $G_1 times G_2$, then $V$ is isomorphic to $V_1 boxtimes V_2$ for some irreducible representations $V_1$ and $V_2$ of $G_1$ and $G_2$ respectively. This means that if we know the representations of $G_1$ and $G_2$, then using the $boxtimes$ construction we can get to all the (irreducible) representations of $G_1 times G_2$. Conversely, the $boxtimes$ product of two irreducible representations always produces an irreducible representation of $G_1 times G_2$.



          On the other hand, $V_1 boxplus V_2$ is always reducible as a $G_1 times G_2$ representation, since both vector subspaces $V_1$ and $V_2$ are stable under the $G_1 times G_2$ action. On the $V_1$ subspace, really only the $G_1$ part of the group acts, and the $G_2$ part acts trivially, and similarly for the $V_2$ subspace. We cannot produce all irreducible representations of $G_1 times G_2$ using this construction, which can already be seen in the example $G_1 = G_2 = mathbb{Z} / 2 mathbb{Z}$.






          share|cite|improve this answer









          $endgroup$













            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "69"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3140677%2fdirect-sum-of-representation-of-product-groups%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            6












            $begingroup$

            When $V_1$ and $V_2$ are representations of $G_1$ and $G_2$ respectively, I'll use $V_1 boxtimes V_2$ to mean the representation of $G_1 times G_2$ with underlying vector space $V_1 otimes_{mathbb{C}} V_2$, and $V_1 boxplus V_2$ to mean the representation of $G_1 times G_2$ with underlying vector space $V_1 oplus V_2$.



            If $V$ is an irreducible representation of $G_1 times G_2$, then $V$ is isomorphic to $V_1 boxtimes V_2$ for some irreducible representations $V_1$ and $V_2$ of $G_1$ and $G_2$ respectively. This means that if we know the representations of $G_1$ and $G_2$, then using the $boxtimes$ construction we can get to all the (irreducible) representations of $G_1 times G_2$. Conversely, the $boxtimes$ product of two irreducible representations always produces an irreducible representation of $G_1 times G_2$.



            On the other hand, $V_1 boxplus V_2$ is always reducible as a $G_1 times G_2$ representation, since both vector subspaces $V_1$ and $V_2$ are stable under the $G_1 times G_2$ action. On the $V_1$ subspace, really only the $G_1$ part of the group acts, and the $G_2$ part acts trivially, and similarly for the $V_2$ subspace. We cannot produce all irreducible representations of $G_1 times G_2$ using this construction, which can already be seen in the example $G_1 = G_2 = mathbb{Z} / 2 mathbb{Z}$.






            share|cite|improve this answer









            $endgroup$


















              6












              $begingroup$

              When $V_1$ and $V_2$ are representations of $G_1$ and $G_2$ respectively, I'll use $V_1 boxtimes V_2$ to mean the representation of $G_1 times G_2$ with underlying vector space $V_1 otimes_{mathbb{C}} V_2$, and $V_1 boxplus V_2$ to mean the representation of $G_1 times G_2$ with underlying vector space $V_1 oplus V_2$.



              If $V$ is an irreducible representation of $G_1 times G_2$, then $V$ is isomorphic to $V_1 boxtimes V_2$ for some irreducible representations $V_1$ and $V_2$ of $G_1$ and $G_2$ respectively. This means that if we know the representations of $G_1$ and $G_2$, then using the $boxtimes$ construction we can get to all the (irreducible) representations of $G_1 times G_2$. Conversely, the $boxtimes$ product of two irreducible representations always produces an irreducible representation of $G_1 times G_2$.



              On the other hand, $V_1 boxplus V_2$ is always reducible as a $G_1 times G_2$ representation, since both vector subspaces $V_1$ and $V_2$ are stable under the $G_1 times G_2$ action. On the $V_1$ subspace, really only the $G_1$ part of the group acts, and the $G_2$ part acts trivially, and similarly for the $V_2$ subspace. We cannot produce all irreducible representations of $G_1 times G_2$ using this construction, which can already be seen in the example $G_1 = G_2 = mathbb{Z} / 2 mathbb{Z}$.






              share|cite|improve this answer









              $endgroup$
















                6












                6








                6





                $begingroup$

                When $V_1$ and $V_2$ are representations of $G_1$ and $G_2$ respectively, I'll use $V_1 boxtimes V_2$ to mean the representation of $G_1 times G_2$ with underlying vector space $V_1 otimes_{mathbb{C}} V_2$, and $V_1 boxplus V_2$ to mean the representation of $G_1 times G_2$ with underlying vector space $V_1 oplus V_2$.



                If $V$ is an irreducible representation of $G_1 times G_2$, then $V$ is isomorphic to $V_1 boxtimes V_2$ for some irreducible representations $V_1$ and $V_2$ of $G_1$ and $G_2$ respectively. This means that if we know the representations of $G_1$ and $G_2$, then using the $boxtimes$ construction we can get to all the (irreducible) representations of $G_1 times G_2$. Conversely, the $boxtimes$ product of two irreducible representations always produces an irreducible representation of $G_1 times G_2$.



                On the other hand, $V_1 boxplus V_2$ is always reducible as a $G_1 times G_2$ representation, since both vector subspaces $V_1$ and $V_2$ are stable under the $G_1 times G_2$ action. On the $V_1$ subspace, really only the $G_1$ part of the group acts, and the $G_2$ part acts trivially, and similarly for the $V_2$ subspace. We cannot produce all irreducible representations of $G_1 times G_2$ using this construction, which can already be seen in the example $G_1 = G_2 = mathbb{Z} / 2 mathbb{Z}$.






                share|cite|improve this answer









                $endgroup$



                When $V_1$ and $V_2$ are representations of $G_1$ and $G_2$ respectively, I'll use $V_1 boxtimes V_2$ to mean the representation of $G_1 times G_2$ with underlying vector space $V_1 otimes_{mathbb{C}} V_2$, and $V_1 boxplus V_2$ to mean the representation of $G_1 times G_2$ with underlying vector space $V_1 oplus V_2$.



                If $V$ is an irreducible representation of $G_1 times G_2$, then $V$ is isomorphic to $V_1 boxtimes V_2$ for some irreducible representations $V_1$ and $V_2$ of $G_1$ and $G_2$ respectively. This means that if we know the representations of $G_1$ and $G_2$, then using the $boxtimes$ construction we can get to all the (irreducible) representations of $G_1 times G_2$. Conversely, the $boxtimes$ product of two irreducible representations always produces an irreducible representation of $G_1 times G_2$.



                On the other hand, $V_1 boxplus V_2$ is always reducible as a $G_1 times G_2$ representation, since both vector subspaces $V_1$ and $V_2$ are stable under the $G_1 times G_2$ action. On the $V_1$ subspace, really only the $G_1$ part of the group acts, and the $G_2$ part acts trivially, and similarly for the $V_2$ subspace. We cannot produce all irreducible representations of $G_1 times G_2$ using this construction, which can already be seen in the example $G_1 = G_2 = mathbb{Z} / 2 mathbb{Z}$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 4 hours ago









                JoppyJoppy

                5,818421




                5,818421






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3140677%2fdirect-sum-of-representation-of-product-groups%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    Knooppunt Holsloot

                    Altaar (religie)

                    Gregoriusmis