Proof or counterexample for isomorphism of group representationsReference request: Indecomposable representations of posetsUnitary representations of locally compact topological groupsUnitary G-module*-representations, unitary representations, and adjunctionsReducibility of unitary representationsBasis of $SO(n)$-representationsVariational criterion for commuting tori in a compact Lie groupDoes $pi otimes overlinepi$ contain weakly the identity representation $1_G$?Mistake in the proof of Theorem 2.24 of Quiver Representations by Ralf Schiffler?What happens if the trivial representation is the only irreducible representation?

In the late 1940’s to early 1950’s what technology was available that could melt a LOT of ice?

Specifying a starting column with colortbl package and xcolor

Was it really inappropriate to write a pull request for the company I interviewed with?

How to resolve: Reviewer #1 says remove section X vs. Reviewer #2 says expand section X

Expressing logarithmic equations without logs

How can I manipulate the output of Information?

Crossing a border with an infant of a different citizenship

Why couldn't the separatists legally leave the Republic?

Plausibility of Mushroom Buildings

Confusion about Complex Continued Fraction

Why does Central Limit Theorem break down in my simulation?

What do *foreign films* mean for an American?

What's the 'present simple' form of the word "нашла́" in 3rd person singular female?

What stops an assembly program from crashing the operating system?

Called into a meeting and told we are being made redundant (laid off) and "not to share outside". Can I tell my partner?

From an axiomatic set theoric approach why can we take uncountable unions?

Drawing close together horizontal lines in Latex

Why is there an extra space when I type "ls" in the Desktop directory?

Do items de-spawn?

Can I negotiate a patent idea for a raise, under French law?

Signed and unsigned numbers

Conservation of Mass and Energy

Giving a career talk in my old university, how prominently should I tell students my salary?

Recommendation letter by significant other if you worked with them professionally?



Proof or counterexample for isomorphism of group representations


Reference request: Indecomposable representations of posetsUnitary representations of locally compact topological groupsUnitary G-module*-representations, unitary representations, and adjunctionsReducibility of unitary representationsBasis of $SO(n)$-representationsVariational criterion for commuting tori in a compact Lie groupDoes $pi otimes overlinepi$ contain weakly the identity representation $1_G$?Mistake in the proof of Theorem 2.24 of Quiver Representations by Ralf Schiffler?What happens if the trivial representation is the only irreducible representation?













1












$begingroup$


Let $(pi , V_pi)$ be an irreducible unitary representation of the (locally compact) group $G$. Let $V_pi^n = V_pi oplus ldots oplus V_pi$ be the $n$-fold direct sum of $V_pi$ on which we have the (unitary) representation $pi^n$ where $$pi^n(g) big(v_1 , ldots , v_n big) := big(pi(g) v_1 , ldots , pi(g) v_n big) .$$
I strongly suspect, that for fixed $v in V_pi^n$ with $v_1 neq 0$ the map $$operatornamePr colon V_pi^n to V_pi colon (v_1 , ldots , v_n ) longmapsto v_1$$is a $G$-intertwining isomorphism bewtween$$left( pi^n , overlinelangle pi(g) v ~ : g in G rangle right) longleftrightarrow big( pi , V_pi big) .$$
The $G$-intertwining property as well as surjectivity are clear but I'm having trouble showing injectivity. I'm grateful for any hints as well as for a counterexample, if there is one. If this makes it any easier, the solution to that problem would be sufficient for the case $G = operatornameSL_2 big(mathbbRbig)$.










share|cite|improve this question









$endgroup$
















    1












    $begingroup$


    Let $(pi , V_pi)$ be an irreducible unitary representation of the (locally compact) group $G$. Let $V_pi^n = V_pi oplus ldots oplus V_pi$ be the $n$-fold direct sum of $V_pi$ on which we have the (unitary) representation $pi^n$ where $$pi^n(g) big(v_1 , ldots , v_n big) := big(pi(g) v_1 , ldots , pi(g) v_n big) .$$
    I strongly suspect, that for fixed $v in V_pi^n$ with $v_1 neq 0$ the map $$operatornamePr colon V_pi^n to V_pi colon (v_1 , ldots , v_n ) longmapsto v_1$$is a $G$-intertwining isomorphism bewtween$$left( pi^n , overlinelangle pi(g) v ~ : g in G rangle right) longleftrightarrow big( pi , V_pi big) .$$
    The $G$-intertwining property as well as surjectivity are clear but I'm having trouble showing injectivity. I'm grateful for any hints as well as for a counterexample, if there is one. If this makes it any easier, the solution to that problem would be sufficient for the case $G = operatornameSL_2 big(mathbbRbig)$.










    share|cite|improve this question









    $endgroup$














      1












      1








      1





      $begingroup$


      Let $(pi , V_pi)$ be an irreducible unitary representation of the (locally compact) group $G$. Let $V_pi^n = V_pi oplus ldots oplus V_pi$ be the $n$-fold direct sum of $V_pi$ on which we have the (unitary) representation $pi^n$ where $$pi^n(g) big(v_1 , ldots , v_n big) := big(pi(g) v_1 , ldots , pi(g) v_n big) .$$
      I strongly suspect, that for fixed $v in V_pi^n$ with $v_1 neq 0$ the map $$operatornamePr colon V_pi^n to V_pi colon (v_1 , ldots , v_n ) longmapsto v_1$$is a $G$-intertwining isomorphism bewtween$$left( pi^n , overlinelangle pi(g) v ~ : g in G rangle right) longleftrightarrow big( pi , V_pi big) .$$
      The $G$-intertwining property as well as surjectivity are clear but I'm having trouble showing injectivity. I'm grateful for any hints as well as for a counterexample, if there is one. If this makes it any easier, the solution to that problem would be sufficient for the case $G = operatornameSL_2 big(mathbbRbig)$.










      share|cite|improve this question









      $endgroup$




      Let $(pi , V_pi)$ be an irreducible unitary representation of the (locally compact) group $G$. Let $V_pi^n = V_pi oplus ldots oplus V_pi$ be the $n$-fold direct sum of $V_pi$ on which we have the (unitary) representation $pi^n$ where $$pi^n(g) big(v_1 , ldots , v_n big) := big(pi(g) v_1 , ldots , pi(g) v_n big) .$$
      I strongly suspect, that for fixed $v in V_pi^n$ with $v_1 neq 0$ the map $$operatornamePr colon V_pi^n to V_pi colon (v_1 , ldots , v_n ) longmapsto v_1$$is a $G$-intertwining isomorphism bewtween$$left( pi^n , overlinelangle pi(g) v ~ : g in G rangle right) longleftrightarrow big( pi , V_pi big) .$$
      The $G$-intertwining property as well as surjectivity are clear but I'm having trouble showing injectivity. I'm grateful for any hints as well as for a counterexample, if there is one. If this makes it any easier, the solution to that problem would be sufficient for the case $G = operatornameSL_2 big(mathbbRbig)$.







      representation-theory lie-groups vector-space-isomorphism






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked yesterday









      TargonTargon

      535




      535




















          1 Answer
          1






          active

          oldest

          votes


















          0












          $begingroup$

          It's not true in general, even for finite dimensional representations of finite groups.



          Take for instance $mathfrakS_3$ acting on $mathbbR^3$, take the irreducible subrepresentation $V=mathbfxin mathbbR^3 mid sum x_i = 0$.



          Then take $v =(v_1,v_2) in Voplus V$ with $v_1 = (1,1,-2)$, $v_2 = (1,2,-3)$.



          Then with $sigma = (1 2)$ you have $sigma v - v$ which is sent to $sigma v_1 - v_1=0$ but $sigma v - v = (sigma v_1 - v_1, sigma v_2 - v_2) = (0, (1,-1,0))neq 0$



          (To construct the counterexample I thought of the easiest way for $displaystylesum_g lambda_g gv_1 = 0$ but $displaystylesum_g lambda_g g$ not $0$ on $V$, then it suffices to find $v_2$ not in the kernel; for this I looked for examples of $g-1$ not being $0$ (that is, $g$ is not in the kernel of the representation), but having a zero (that is, $g$ having a fixed point) - this led me to my example)






          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%2f3141072%2fproof-or-counterexample-for-isomorphism-of-group-representations%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









            0












            $begingroup$

            It's not true in general, even for finite dimensional representations of finite groups.



            Take for instance $mathfrakS_3$ acting on $mathbbR^3$, take the irreducible subrepresentation $V=mathbfxin mathbbR^3 mid sum x_i = 0$.



            Then take $v =(v_1,v_2) in Voplus V$ with $v_1 = (1,1,-2)$, $v_2 = (1,2,-3)$.



            Then with $sigma = (1 2)$ you have $sigma v - v$ which is sent to $sigma v_1 - v_1=0$ but $sigma v - v = (sigma v_1 - v_1, sigma v_2 - v_2) = (0, (1,-1,0))neq 0$



            (To construct the counterexample I thought of the easiest way for $displaystylesum_g lambda_g gv_1 = 0$ but $displaystylesum_g lambda_g g$ not $0$ on $V$, then it suffices to find $v_2$ not in the kernel; for this I looked for examples of $g-1$ not being $0$ (that is, $g$ is not in the kernel of the representation), but having a zero (that is, $g$ having a fixed point) - this led me to my example)






            share|cite|improve this answer









            $endgroup$

















              0












              $begingroup$

              It's not true in general, even for finite dimensional representations of finite groups.



              Take for instance $mathfrakS_3$ acting on $mathbbR^3$, take the irreducible subrepresentation $V=mathbfxin mathbbR^3 mid sum x_i = 0$.



              Then take $v =(v_1,v_2) in Voplus V$ with $v_1 = (1,1,-2)$, $v_2 = (1,2,-3)$.



              Then with $sigma = (1 2)$ you have $sigma v - v$ which is sent to $sigma v_1 - v_1=0$ but $sigma v - v = (sigma v_1 - v_1, sigma v_2 - v_2) = (0, (1,-1,0))neq 0$



              (To construct the counterexample I thought of the easiest way for $displaystylesum_g lambda_g gv_1 = 0$ but $displaystylesum_g lambda_g g$ not $0$ on $V$, then it suffices to find $v_2$ not in the kernel; for this I looked for examples of $g-1$ not being $0$ (that is, $g$ is not in the kernel of the representation), but having a zero (that is, $g$ having a fixed point) - this led me to my example)






              share|cite|improve this answer









              $endgroup$















                0












                0








                0





                $begingroup$

                It's not true in general, even for finite dimensional representations of finite groups.



                Take for instance $mathfrakS_3$ acting on $mathbbR^3$, take the irreducible subrepresentation $V=mathbfxin mathbbR^3 mid sum x_i = 0$.



                Then take $v =(v_1,v_2) in Voplus V$ with $v_1 = (1,1,-2)$, $v_2 = (1,2,-3)$.



                Then with $sigma = (1 2)$ you have $sigma v - v$ which is sent to $sigma v_1 - v_1=0$ but $sigma v - v = (sigma v_1 - v_1, sigma v_2 - v_2) = (0, (1,-1,0))neq 0$



                (To construct the counterexample I thought of the easiest way for $displaystylesum_g lambda_g gv_1 = 0$ but $displaystylesum_g lambda_g g$ not $0$ on $V$, then it suffices to find $v_2$ not in the kernel; for this I looked for examples of $g-1$ not being $0$ (that is, $g$ is not in the kernel of the representation), but having a zero (that is, $g$ having a fixed point) - this led me to my example)






                share|cite|improve this answer









                $endgroup$



                It's not true in general, even for finite dimensional representations of finite groups.



                Take for instance $mathfrakS_3$ acting on $mathbbR^3$, take the irreducible subrepresentation $V=mathbfxin mathbbR^3 mid sum x_i = 0$.



                Then take $v =(v_1,v_2) in Voplus V$ with $v_1 = (1,1,-2)$, $v_2 = (1,2,-3)$.



                Then with $sigma = (1 2)$ you have $sigma v - v$ which is sent to $sigma v_1 - v_1=0$ but $sigma v - v = (sigma v_1 - v_1, sigma v_2 - v_2) = (0, (1,-1,0))neq 0$



                (To construct the counterexample I thought of the easiest way for $displaystylesum_g lambda_g gv_1 = 0$ but $displaystylesum_g lambda_g g$ not $0$ on $V$, then it suffices to find $v_2$ not in the kernel; for this I looked for examples of $g-1$ not being $0$ (that is, $g$ is not in the kernel of the representation), but having a zero (that is, $g$ having a fixed point) - this led me to my example)







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered yesterday









                MaxMax

                15.1k11143




                15.1k11143



























                    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%2f3141072%2fproof-or-counterexample-for-isomorphism-of-group-representations%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

                    How should I support this large drywall patch? Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?How do I cover large gaps in drywall?How do I keep drywall around a patch from crumbling?Can I glue a second layer of drywall?How to patch long strip on drywall?Large drywall patch: how to avoid bulging seams?Drywall Mesh Patch vs. Bulge? To remove or not to remove?How to fix this drywall job?Prep drywall before backsplashWhat's the best way to fix this horrible drywall patch job?Drywall patching using 3M Patch Plus Primer

                    random experiment with two different functions on unit interval Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Random variable and probability space notionsRandom Walk with EdgesFinding functions where the increase over a random interval is Poisson distributedNumber of days until dayCan an observed event in fact be of zero probability?Unit random processmodels of coins and uniform distributionHow to get the number of successes given $n$ trials , probability $P$ and a random variable $X$Absorbing Markov chain in a computer. Is “almost every” turned into always convergence in computer executions?Stopped random walk is not uniformly integrable

                    Lowndes Grove History Architecture References Navigation menu32°48′6″N 79°57′58″W / 32.80167°N 79.96611°W / 32.80167; -79.9661132°48′6″N 79°57′58″W / 32.80167°N 79.96611°W / 32.80167; -79.9661178002500"National Register Information System"Historic houses of South Carolina"Lowndes Grove""+32° 48' 6.00", −79° 57' 58.00""Lowndes Grove, Charleston County (260 St. Margaret St., Charleston)""Lowndes Grove"The Charleston ExpositionIt Happened in South Carolina"Lowndes Grove (House), Saint Margaret Street & Sixth Avenue, Charleston, Charleston County, SC(Photographs)"Plantations of the Carolina Low Countrye