Fundamental representations and weight space dimension











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For the Lie algebra $frak{sl}_n$, 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?










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  • Some unpubllished notes from 2014 and 2016 are on my webpage, with further bibliographic references: people.math.umass.edu/~jeh/pub/fund.pdf.
    – Jim Humphreys
    Nov 5 at 15:59

















up vote
4
down vote

favorite












For the Lie algebra $frak{sl}_n$, 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?










share|cite|improve this question






















  • Some unpubllished notes from 2014 and 2016 are on my webpage, with further bibliographic references: people.math.umass.edu/~jeh/pub/fund.pdf.
    – Jim Humphreys
    Nov 5 at 15:59















up vote
4
down vote

favorite









up vote
4
down vote

favorite











For the Lie algebra $frak{sl}_n$, 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?










share|cite|improve this question













For the Lie algebra $frak{sl}_n$, 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






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asked Nov 5 at 12:29









Pierre Dubois

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  • Some unpubllished notes from 2014 and 2016 are on my webpage, with further bibliographic references: people.math.umass.edu/~jeh/pub/fund.pdf.
    – Jim Humphreys
    Nov 5 at 15:59




















  • Some unpubllished notes from 2014 and 2016 are on my webpage, with further bibliographic references: people.math.umass.edu/~jeh/pub/fund.pdf.
    – Jim Humphreys
    Nov 5 at 15:59


















Some unpubllished notes from 2014 and 2016 are on my webpage, with further bibliographic references: people.math.umass.edu/~jeh/pub/fund.pdf.
– Jim Humphreys
Nov 5 at 15:59






Some unpubllished notes from 2014 and 2016 are on my webpage, with further bibliographic references: people.math.umass.edu/~jeh/pub/fund.pdf.
– Jim Humphreys
Nov 5 at 15:59












2 Answers
2






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up vote
5
down vote



accepted










Let $mathfrak{g}$ 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 $mathfrak{g}$ 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 $B_l$ ($l geq 2$): $lambda = varpi_1$ or $varpi_l$.

  • Type $C_l$ ($l geq 3$): $lambda = varpi_1$. For type $C_3$, also $lambda = varpi_3$.

  • 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 $0 leq d < p$).






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  • 1




    It's also worth pointing out that outside types A and C, the adjoint representation is fundamental, but the $0$ weight space (i.e. a Cartan subalgebra) is not one-dimensional.
    – Paul Levy
    Nov 7 at 10:23


















up vote
3
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
    Nov 5 at 15:03










  • For type $B$, the natural representation has all weight spaces of dimension $1$, but it is not minuscule. Another example would be irreducible of highest weight $varpi_3$ for type $C_3$.
    – Mikko Korhonen
    Nov 6 at 1:47












  • @MikkoKorhonen Oops. Thanks for spotting that mistake.
    – Vít Tuček
    Nov 6 at 8:39










  • @PierreDubois Sorry, I don't really know much about these. If you happen to find some nice reference, please let me know.
    – Vít Tuček
    Nov 6 at 8:40











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2 Answers
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2 Answers
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up vote
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Let $mathfrak{g}$ 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 $mathfrak{g}$ 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 $B_l$ ($l geq 2$): $lambda = varpi_1$ or $varpi_l$.

  • Type $C_l$ ($l geq 3$): $lambda = varpi_1$. For type $C_3$, also $lambda = varpi_3$.

  • 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



















  • 1




    It's also worth pointing out that outside types A and C, the adjoint representation is fundamental, but the $0$ weight space (i.e. a Cartan subalgebra) is not one-dimensional.
    – Paul Levy
    Nov 7 at 10:23















up vote
5
down vote



accepted










Let $mathfrak{g}$ 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 $mathfrak{g}$ 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 $B_l$ ($l geq 2$): $lambda = varpi_1$ or $varpi_l$.

  • Type $C_l$ ($l geq 3$): $lambda = varpi_1$. For type $C_3$, also $lambda = varpi_3$.

  • 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



















  • 1




    It's also worth pointing out that outside types A and C, the adjoint representation is fundamental, but the $0$ weight space (i.e. a Cartan subalgebra) is not one-dimensional.
    – Paul Levy
    Nov 7 at 10:23













up vote
5
down vote



accepted







up vote
5
down vote



accepted






Let $mathfrak{g}$ 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 $mathfrak{g}$ 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 $B_l$ ($l geq 2$): $lambda = varpi_1$ or $varpi_l$.

  • Type $C_l$ ($l geq 3$): $lambda = varpi_1$. For type $C_3$, also $lambda = varpi_3$.

  • 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 $mathfrak{g}$ 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 $mathfrak{g}$ 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 $B_l$ ($l geq 2$): $lambda = varpi_1$ or $varpi_l$.

  • Type $C_l$ ($l geq 3$): $lambda = varpi_1$. For type $C_3$, also $lambda = varpi_3$.

  • 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








edited Nov 6 at 2:00

























answered Nov 5 at 14:14









Mikko Korhonen

978814




978814








  • 1




    It's also worth pointing out that outside types A and C, the adjoint representation is fundamental, but the $0$ weight space (i.e. a Cartan subalgebra) is not one-dimensional.
    – Paul Levy
    Nov 7 at 10:23














  • 1




    It's also worth pointing out that outside types A and C, the adjoint representation is fundamental, but the $0$ weight space (i.e. a Cartan subalgebra) is not one-dimensional.
    – Paul Levy
    Nov 7 at 10:23








1




1




It's also worth pointing out that outside types A and C, the adjoint representation is fundamental, but the $0$ weight space (i.e. a Cartan subalgebra) is not one-dimensional.
– Paul Levy
Nov 7 at 10:23




It's also worth pointing out that outside types A and C, the adjoint representation is fundamental, but the $0$ weight space (i.e. a Cartan subalgebra) is not one-dimensional.
– Paul Levy
Nov 7 at 10:23










up vote
3
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
    Nov 5 at 15:03










  • For type $B$, the natural representation has all weight spaces of dimension $1$, but it is not minuscule. Another example would be irreducible of highest weight $varpi_3$ for type $C_3$.
    – Mikko Korhonen
    Nov 6 at 1:47












  • @MikkoKorhonen Oops. Thanks for spotting that mistake.
    – Vít Tuček
    Nov 6 at 8:39










  • @PierreDubois Sorry, I don't really know much about these. If you happen to find some nice reference, please let me know.
    – Vít Tuček
    Nov 6 at 8:40















up vote
3
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
    Nov 5 at 15:03










  • For type $B$, the natural representation has all weight spaces of dimension $1$, but it is not minuscule. Another example would be irreducible of highest weight $varpi_3$ for type $C_3$.
    – Mikko Korhonen
    Nov 6 at 1:47












  • @MikkoKorhonen Oops. Thanks for spotting that mistake.
    – Vít Tuček
    Nov 6 at 8:39










  • @PierreDubois Sorry, I don't really know much about these. If you happen to find some nice reference, please let me know.
    – Vít Tuček
    Nov 6 at 8:40













up vote
3
down vote










up vote
3
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








edited Nov 6 at 8:39

























answered Nov 5 at 14:17









Vít Tuček

4,75411747




4,75411747












  • Thanks for the terminology. I have also heard people talk about cominiscule representations . . . I guess these are related?
    – Pierre Dubois
    Nov 5 at 15:03










  • For type $B$, the natural representation has all weight spaces of dimension $1$, but it is not minuscule. Another example would be irreducible of highest weight $varpi_3$ for type $C_3$.
    – Mikko Korhonen
    Nov 6 at 1:47












  • @MikkoKorhonen Oops. Thanks for spotting that mistake.
    – Vít Tuček
    Nov 6 at 8:39










  • @PierreDubois Sorry, I don't really know much about these. If you happen to find some nice reference, please let me know.
    – Vít Tuček
    Nov 6 at 8:40


















  • Thanks for the terminology. I have also heard people talk about cominiscule representations . . . I guess these are related?
    – Pierre Dubois
    Nov 5 at 15:03










  • For type $B$, the natural representation has all weight spaces of dimension $1$, but it is not minuscule. Another example would be irreducible of highest weight $varpi_3$ for type $C_3$.
    – Mikko Korhonen
    Nov 6 at 1:47












  • @MikkoKorhonen Oops. Thanks for spotting that mistake.
    – Vít Tuček
    Nov 6 at 8:39










  • @PierreDubois Sorry, I don't really know much about these. If you happen to find some nice reference, please let me know.
    – Vít Tuček
    Nov 6 at 8:40
















Thanks for the terminology. I have also heard people talk about cominiscule representations . . . I guess these are related?
– Pierre Dubois
Nov 5 at 15:03




Thanks for the terminology. I have also heard people talk about cominiscule representations . . . I guess these are related?
– Pierre Dubois
Nov 5 at 15:03












For type $B$, the natural representation has all weight spaces of dimension $1$, but it is not minuscule. Another example would be irreducible of highest weight $varpi_3$ for type $C_3$.
– Mikko Korhonen
Nov 6 at 1:47






For type $B$, the natural representation has all weight spaces of dimension $1$, but it is not minuscule. Another example would be irreducible of highest weight $varpi_3$ for type $C_3$.
– Mikko Korhonen
Nov 6 at 1:47














@MikkoKorhonen Oops. Thanks for spotting that mistake.
– Vít Tuček
Nov 6 at 8:39




@MikkoKorhonen Oops. Thanks for spotting that mistake.
– Vít Tuček
Nov 6 at 8:39












@PierreDubois Sorry, I don't really know much about these. If you happen to find some nice reference, please let me know.
– Vít Tuček
Nov 6 at 8:40




@PierreDubois Sorry, I don't really know much about these. If you happen to find some nice reference, please let me know.
– Vít Tuček
Nov 6 at 8:40


















 

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