Fundamental representations and weight space dimension

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
2
down vote

favorite












For the Lie algebra $fraksl_n$, its fundamental representations can be realised as the exterior powers of the first fundamental representation. From this we can see that their weight spaces are all $1$-dimensional. Is this true for the other series - are the weight spaces of the fundamental representations always $1$-dimensional. If this is true, does it identify the fundamental representations, that is are the fundaental representation precisely those with $1$-dimensional weight spaces?






rt.representation-theory lie-groups lie-algebras







share|cite|improve this question

























    up vote
    2
    down vote

    favorite












    For the Lie algebra $fraksl_n$, its fundamental representations can be realised as the exterior powers of the first fundamental representation. From this we can see that their weight spaces are all $1$-dimensional. Is this true for the other series - are the weight spaces of the fundamental representations always $1$-dimensional. If this is true, does it identify the fundamental representations, that is are the fundaental representation precisely those with $1$-dimensional weight spaces?






    rt.representation-theory lie-groups lie-algebras







    share|cite|improve this question























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      For the Lie algebra $fraksl_n$, its fundamental representations can be realised as the exterior powers of the first fundamental representation. From this we can see that their weight spaces are all $1$-dimensional. Is this true for the other series - are the weight spaces of the fundamental representations always $1$-dimensional. If this is true, does it identify the fundamental representations, that is are the fundaental representation precisely those with $1$-dimensional weight spaces?






      rt.representation-theory lie-groups lie-algebras







      share|cite|improve this question













      For the Lie algebra $fraksl_n$, its fundamental representations can be realised as the exterior powers of the first fundamental representation. From this we can see that their weight spaces are all $1$-dimensional. Is this true for the other series - are the weight spaces of the fundamental representations always $1$-dimensional. If this is true, does it identify the fundamental representations, that is are the fundaental representation precisely those with $1$-dimensional weight spaces?






      rt.representation-theory lie-groups lie-algebras




      rt.representation-theory lie-groups lie-algebras






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 2 hours ago









      Pierre Dubois

      875




      875




















          2 Answers
          2






          active

          oldest

          votes

















          up vote
          2
          down vote



          accepted










          Let $mathfrakg$ be a simple Lie algebra over an algebraically closed field of characteristic $0$ and denote the fundamental highest weights by $varpi_1, ldots, varpi_l$, where $l$ is the rank of $mathfrakg$ and the ordering of the $varpi_i$ is the usual one (Bourbaki).



          The cases where an irreducible representation of highest weight $lambda$ has all weight spaces $1$-dimensional are the following:



          • Type $A_l$: $lambda = varpi_i, c varpi_1$, or $cvarpi_l$.

          • Type $B_l$: $lambda = varpi_1$ or $varpi_l$.

          • Type $C_l$: $lambda = varpi_1$ or $varpi_l$.

          • Type $D_l$ ($l geq 4$): $lambda = varpi_1$, $varpi_l-1$ or $varpi_l$.

          • Type $G_2$: $lambda = varpi_1$.

          • Type $E_6$: $lambda = varpi_1$ or $varpi_6$.

          • Type $E_7$: $lambda = varpi_7$.

          So most fundamental representations do not have $1$-dimensional weight spaces. Also, in type $A$ you have non-fundamental representations which have all weight spaces $1$-dimensional. These can be realized as symmetric powers of the natural irreducible representation and its dual.



          For a reference, see Chapter 6 in Seitz, The maximal subgroups of classical algebraic groups (Memoirs of the AMS). Seitz works with algebraic groups, but this will of course give you the result for Lie algebras as well.



          Seitz actually proves a result over an algebraically closed field of characteristic $p geq 0$. In characteristic $p > 0$ we get some additional examples, such as $lambda = d varpi_i + (p-d-1) varpi_i+1$ in type $A_l$ (for $1 leq i < l$ and $0 leq d < p$).






          share|cite|improve this answer



























            up vote
            1
            down vote













            Such representations are actually quite rare and can be found under the name minuscule representations.






            share|cite|improve this answer




















            • Thanks for the terminology. I have also heard people talk about cominiscule representations . . . I guess these are related?
              – Pierre Dubois
              9 mins ago










            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: "504"
            ;
            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',
            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%2fmathoverflow.net%2fquestions%2f314597%2ffundamental-representations-and-weight-space-dimension%23new-answer', 'question_page');

            );

            Post as a guest

















            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            2
            down vote



            accepted










            Let $mathfrakg$ be a simple Lie algebra over an algebraically closed field of characteristic $0$ and denote the fundamental highest weights by $varpi_1, ldots, varpi_l$, where $l$ is the rank of $mathfrakg$ and the ordering of the $varpi_i$ is the usual one (Bourbaki).



            The cases where an irreducible representation of highest weight $lambda$ has all weight spaces $1$-dimensional are the following:



            • Type $A_l$: $lambda = varpi_i, c varpi_1$, or $cvarpi_l$.

            • Type $B_l$: $lambda = varpi_1$ or $varpi_l$.

            • Type $C_l$: $lambda = varpi_1$ or $varpi_l$.

            • Type $D_l$ ($l geq 4$): $lambda = varpi_1$, $varpi_l-1$ or $varpi_l$.

            • Type $G_2$: $lambda = varpi_1$.

            • Type $E_6$: $lambda = varpi_1$ or $varpi_6$.

            • Type $E_7$: $lambda = varpi_7$.

            So most fundamental representations do not have $1$-dimensional weight spaces. Also, in type $A$ you have non-fundamental representations which have all weight spaces $1$-dimensional. These can be realized as symmetric powers of the natural irreducible representation and its dual.



            For a reference, see Chapter 6 in Seitz, The maximal subgroups of classical algebraic groups (Memoirs of the AMS). Seitz works with algebraic groups, but this will of course give you the result for Lie algebras as well.



            Seitz actually proves a result over an algebraically closed field of characteristic $p geq 0$. In characteristic $p > 0$ we get some additional examples, such as $lambda = d varpi_i + (p-d-1) varpi_i+1$ in type $A_l$ (for $1 leq i < l$ and $0 leq d < p$).






            share|cite|improve this answer
























              up vote
              2
              down vote



              accepted










              Let $mathfrakg$ be a simple Lie algebra over an algebraically closed field of characteristic $0$ and denote the fundamental highest weights by $varpi_1, ldots, varpi_l$, where $l$ is the rank of $mathfrakg$ and the ordering of the $varpi_i$ is the usual one (Bourbaki).



              The cases where an irreducible representation of highest weight $lambda$ has all weight spaces $1$-dimensional are the following:



              • Type $A_l$: $lambda = varpi_i, c varpi_1$, or $cvarpi_l$.

              • Type $B_l$: $lambda = varpi_1$ or $varpi_l$.

              • Type $C_l$: $lambda = varpi_1$ or $varpi_l$.

              • Type $D_l$ ($l geq 4$): $lambda = varpi_1$, $varpi_l-1$ or $varpi_l$.

              • Type $G_2$: $lambda = varpi_1$.

              • Type $E_6$: $lambda = varpi_1$ or $varpi_6$.

              • Type $E_7$: $lambda = varpi_7$.

              So most fundamental representations do not have $1$-dimensional weight spaces. Also, in type $A$ you have non-fundamental representations which have all weight spaces $1$-dimensional. These can be realized as symmetric powers of the natural irreducible representation and its dual.



              For a reference, see Chapter 6 in Seitz, The maximal subgroups of classical algebraic groups (Memoirs of the AMS). Seitz works with algebraic groups, but this will of course give you the result for Lie algebras as well.



              Seitz actually proves a result over an algebraically closed field of characteristic $p geq 0$. In characteristic $p > 0$ we get some additional examples, such as $lambda = d varpi_i + (p-d-1) varpi_i+1$ in type $A_l$ (for $1 leq i < l$ and $0 leq d < p$).






              share|cite|improve this answer






















                up vote
                2
                down vote



                accepted







                up vote
                2
                down vote



                accepted






                Let $mathfrakg$ be a simple Lie algebra over an algebraically closed field of characteristic $0$ and denote the fundamental highest weights by $varpi_1, ldots, varpi_l$, where $l$ is the rank of $mathfrakg$ and the ordering of the $varpi_i$ is the usual one (Bourbaki).



                The cases where an irreducible representation of highest weight $lambda$ has all weight spaces $1$-dimensional are the following:



                • Type $A_l$: $lambda = varpi_i, c varpi_1$, or $cvarpi_l$.

                • Type $B_l$: $lambda = varpi_1$ or $varpi_l$.

                • Type $C_l$: $lambda = varpi_1$ or $varpi_l$.

                • Type $D_l$ ($l geq 4$): $lambda = varpi_1$, $varpi_l-1$ or $varpi_l$.

                • Type $G_2$: $lambda = varpi_1$.

                • Type $E_6$: $lambda = varpi_1$ or $varpi_6$.

                • Type $E_7$: $lambda = varpi_7$.

                So most fundamental representations do not have $1$-dimensional weight spaces. Also, in type $A$ you have non-fundamental representations which have all weight spaces $1$-dimensional. These can be realized as symmetric powers of the natural irreducible representation and its dual.



                For a reference, see Chapter 6 in Seitz, The maximal subgroups of classical algebraic groups (Memoirs of the AMS). Seitz works with algebraic groups, but this will of course give you the result for Lie algebras as well.



                Seitz actually proves a result over an algebraically closed field of characteristic $p geq 0$. In characteristic $p > 0$ we get some additional examples, such as $lambda = d varpi_i + (p-d-1) varpi_i+1$ in type $A_l$ (for $1 leq i < l$ and $0 leq d < p$).






                share|cite|improve this answer












                Let $mathfrakg$ be a simple Lie algebra over an algebraically closed field of characteristic $0$ and denote the fundamental highest weights by $varpi_1, ldots, varpi_l$, where $l$ is the rank of $mathfrakg$ and the ordering of the $varpi_i$ is the usual one (Bourbaki).



                The cases where an irreducible representation of highest weight $lambda$ has all weight spaces $1$-dimensional are the following:



                • Type $A_l$: $lambda = varpi_i, c varpi_1$, or $cvarpi_l$.

                • Type $B_l$: $lambda = varpi_1$ or $varpi_l$.

                • Type $C_l$: $lambda = varpi_1$ or $varpi_l$.

                • Type $D_l$ ($l geq 4$): $lambda = varpi_1$, $varpi_l-1$ or $varpi_l$.

                • Type $G_2$: $lambda = varpi_1$.

                • Type $E_6$: $lambda = varpi_1$ or $varpi_6$.

                • Type $E_7$: $lambda = varpi_7$.

                So most fundamental representations do not have $1$-dimensional weight spaces. Also, in type $A$ you have non-fundamental representations which have all weight spaces $1$-dimensional. These can be realized as symmetric powers of the natural irreducible representation and its dual.



                For a reference, see Chapter 6 in Seitz, The maximal subgroups of classical algebraic groups (Memoirs of the AMS). Seitz works with algebraic groups, but this will of course give you the result for Lie algebras as well.



                Seitz actually proves a result over an algebraically closed field of characteristic $p geq 0$. In characteristic $p > 0$ we get some additional examples, such as $lambda = d varpi_i + (p-d-1) varpi_i+1$ in type $A_l$ (for $1 leq i < l$ and $0 leq d < p$).







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 59 mins ago









                Mikko Korhonen

                948814




                948814




















                    up vote
                    1
                    down vote













                    Such representations are actually quite rare and can be found under the name minuscule representations.






                    share|cite|improve this answer




















                    • Thanks for the terminology. I have also heard people talk about cominiscule representations . . . I guess these are related?
                      – Pierre Dubois
                      9 mins ago














                    up vote
                    1
                    down vote













                    Such representations are actually quite rare and can be found under the name minuscule representations.






                    share|cite|improve this answer




















                    • Thanks for the terminology. I have also heard people talk about cominiscule representations . . . I guess these are related?
                      – Pierre Dubois
                      9 mins ago












                    up vote
                    1
                    down vote










                    up vote
                    1
                    down vote









                    Such representations are actually quite rare and can be found under the name minuscule representations.






                    share|cite|improve this answer












                    Such representations are actually quite rare and can be found under the name minuscule representations.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 55 mins ago









                    Vít Tuček

                    4,69411746




                    4,69411746











                    • Thanks for the terminology. I have also heard people talk about cominiscule representations . . . I guess these are related?
                      – Pierre Dubois
                      9 mins ago
















                    • Thanks for the terminology. I have also heard people talk about cominiscule representations . . . I guess these are related?
                      – Pierre Dubois
                      9 mins ago















                    Thanks for the terminology. I have also heard people talk about cominiscule representations . . . I guess these are related?
                    – Pierre Dubois
                    9 mins ago




                    Thanks for the terminology. I have also heard people talk about cominiscule representations . . . I guess these are related?
                    – Pierre Dubois
                    9 mins ago

















                     

                    draft saved


                    draft discarded















































                     


                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function ()
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f314597%2ffundamental-representations-and-weight-space-dimension%23new-answer', 'question_page');

                    );

                    Post as a guest
































































                    Comments

                    Post a Comment

                    Popular posts from this blog

                    What does second last employer means? [closed]

                    Image
                    Clash Royale CLAN TAG #URR8PPP .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty margin-bottom:0; up vote -3 down vote favorite Say, for example I have worked in the following companies, Company A => 2010 - 2011 Company B => 2011 - 2012 Company C => 2012 - Till Date Company D => Have offer Now assume Company D is asking to submit documents from my second last employer. So from the list of companies given, which company would be qualified as second last employer? software-industry employer share | improve this question asked Aug 10 '14 at 5:16 Rajaprabhu Aravindasamy 559 1 6 13 closed as off-topic by Jim G., jmort253 ♦ Aug 11 '14 at 1:58 This question appears to be off-topic. The users who voted to close gave this specific reason: "Questions seeking advice on company-specific regulations, agreements, or policies should be directed to your manager or HR department. Questions that address...
                    Read more

                    List of Gilmore Girls characters

                    Image
                    The season three cast—(Back row): Lane, Michel, Paris, Emily, Richard, Sookie, Miss Patty, Kirk; Front row: Jess, Luke, Lorelai, Rory, Dean This is a list of characters for the comedy-drama television series Gilmore Girls . Contents 1 Main characters 1.1 Lorelai Gilmore 1.2 Rory Gilmore 1.3 Sookie St. James 1.4 Lane Kim 1.5 Michel Gerard 1.6 Luke Danes 1.7 Emily Gilmore 1.8 Richard Gilmore 1.9 Paris Geller 1.10 Dean Forester 1.11 Jess Mariano 1.12 Kirk Gleason 1.13 Jason Stiles 1.14 Logan Huntzberger 2 Major recurring characters 2.1 Christopher Hayden 2.2 Jackson Belleville 2.3 Mrs. Kim 2.4 Tristin DuGray 2.5 Max Medina 2.6 Taylor Doose 2.7 Dave Rygalski 2.8 Marty 3 Recurring characters 3.1 Stars Hollow 3.2 Chilton 3.3 Yale 3.4 Other 4 Notable guest stars 5 Unseen characters 6 Notes 7 References Main characters Actor Character Appearances Season 1 Season 2 Season 3 Season 4 Season 5 Season 6 Season 7 A Year in the Life Lauren Graham Lorelai Gilmore...
                    Read more

                    Confectionery

                    Image
                    Prepared foods made with a lot of sugar or other cabohydrates "Sweetmeat" redirects here. For the racehorse, see Sweetmeat (horse). Not to be confused with Sweetbread. This Kransekake is a traditional Scandinavian baker's confection. Confectionery is the art of making confections , which are food items that are rich in sugar and carbohydrates. Exact definitions are difficult. [1] In general, though, confectionery is divided into two broad and somewhat overlapping categories, bakers' confections and sugar confections. [2] Bakers' confectionery, also called flour confections , includes principally sweet pastries, cakes, and similar baked goods. Sugar confectionery includes candies ( sweets in British English), candied nuts, chocolates, chewing gum, bubble gum, pastillage, and other confections that are made primarily of sugar. In some cases, chocolate confections (confections made of chocolate) are treated as a separate category, as are sugar-free versions of ...
                    Read more