Deciding whether a representation is orthogonal or symplectic The 2019 Stack Overflow...
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Deciding whether a representation is orthogonal or symplectic
The 2019 Stack Overflow Developer Survey Results Are Inirreducible highest weight modulesRepresentations of non-semisimple Lie algebrasWeight spaces of Verma modulesThe relation between Weyl character formula and Frobenius characteristic mapBasis for complex representation with symmetric bilinear formEvery submodule of a cyclic $mathfrak{g}$-module is a weight moduleCriterion for Checking When a Lie Algebra Module is IrreducibleWeight $mathfrak{sl}_2$-module with finite dimensional weight spaces has finite length?Understanding weight spaces of weight module from its composition factors?Irreducible Dual Representation
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I'm trying to understand the proof of Proposition 7 part (iii) from this paper of Dadok https://www.ams.org/journals/tran/1985-288-01/S0002-9947-1985-0773051-1/S0002-9947-1985-0773051-1.pdf (Statement of the proposition on page 128, proof on page 131), but getting stuck on one line in the the middle.
The set-up is as follows: We have an irreducible self-dual (complex) representation $pi_{lambda}:mathfrak{g}to mathfrak{gl}(V)$ of highest weight $lambda$ (say $v_{lambda}in V$ is a heighest weight vector). There is a distinguished set ${beta_1,dots, beta_l}$ of strongly orthogonal positive roots of $mathfrak{g}$ (their specific definition isn't important for this part), and we are assuming that $lambda$ is in the (real) span of the $beta_i$. We are considering the subalgebra $mathfrak{u}subseteq mathfrak{g}$ generated by the rootspaces of the $pm beta_i$ (note $mathfrak{u}cong mathfrak{sl}_2(mathbb{C})^{oplus l}$).
Self-duality of $pi_{lambda}$ implies that there is a non-degenerate invariant bilinear form $B:Vtimes Vto mathbb{C}$, and to determine whether $pi_{lambda}$ is orthogonal or symplectic we need to figure out whether $B$ is symmetric or skew-symmetric. Now he defines $U_{lambda}subseteq V_{lambda}$ as the $U(mathfrak{u})$-submodule generated by the highest weight vector $v_{lambda}$ (the module $U_{lambda}$ will be simple by the theorem of highest weight, using the fact that $lambda$ is in the span of the $beta_i$). The idea of the argument is that he wants to show that one decide whether $B$ is symmetric or skew-symmetric by looking at the restriction $B:U_{lambda}times U_{lambda}to mathbb{C}$. To that end, he writes:
"If $B:Vto Vto mathbb{C}$ is the nondegenerate bilinear form invariant under $pi_{lambda}$, we see that $B$ must remain non-degenerate on $U_{lambda}times U_{lambda}tomathbb{C}$ since the $mathfrak{u}$-module $U_{lambda}$ appears in $V_{lambda}$ with multiplicity one."
Can anyone help me make sense of this line? I can't even figure out what he means by multiplicity in this context.
Thanks!
abstract-algebra modules representation-theory lie-algebras
asked Mar 21 at 14:53
itinerantleoparditinerantleopard
1379
$endgroup$
add a comment |
$begingroup$
I'm trying to understand the proof of Proposition 7 part (iii) from this paper of Dadok https://www.ams.org/journals/tran/1985-288-01/S0002-9947-1985-0773051-1/S0002-9947-1985-0773051-1.pdf (Statement of the proposition on page 128, proof on page 131), but getting stuck on one line in the the middle.
The set-up is as follows: We have an irreducible self-dual (complex) representation $pi_{lambda}:mathfrak{g}to mathfrak{gl}(V)$ of highest weight $lambda$ (say $v_{lambda}in V$ is a heighest weight vector). There is a distinguished set ${beta_1,dots, beta_l}$ of strongly orthogonal positive roots of $mathfrak{g}$ (their specific definition isn't important for this part), and we are assuming that $lambda$ is in the (real) span of the $beta_i$. We are considering the subalgebra $mathfrak{u}subseteq mathfrak{g}$ generated by the rootspaces of the $pm beta_i$ (note $mathfrak{u}cong mathfrak{sl}_2(mathbb{C})^{oplus l}$).
Self-duality of $pi_{lambda}$ implies that there is a non-degenerate invariant bilinear form $B:Vtimes Vto mathbb{C}$, and to determine whether $pi_{lambda}$ is orthogonal or symplectic we need to figure out whether $B$ is symmetric or skew-symmetric. Now he defines $U_{lambda}subseteq V_{lambda}$ as the $U(mathfrak{u})$-submodule generated by the highest weight vector $v_{lambda}$ (the module $U_{lambda}$ will be simple by the theorem of highest weight, using the fact that $lambda$ is in the span of the $beta_i$). The idea of the argument is that he wants to show that one decide whether $B$ is symmetric or skew-symmetric by looking at the restriction $B:U_{lambda}times U_{lambda}to mathbb{C}$. To that end, he writes:
"If $B:Vto Vto mathbb{C}$ is the nondegenerate bilinear form invariant under $pi_{lambda}$, we see that $B$ must remain non-degenerate on $U_{lambda}times U_{lambda}tomathbb{C}$ since the $mathfrak{u}$-module $U_{lambda}$ appears in $V_{lambda}$ with multiplicity one."
Can anyone help me make sense of this line? I can't even figure out what he means by multiplicity in this context.
Thanks!
abstract-algebra modules representation-theory lie-algebras
asked Mar 21 at 14:53
itinerantleoparditinerantleopard
1379
$endgroup$
$begingroup$
Why is $mathfrak{u}congmathfrak{sl}_2$? Isn't it generated by multiple root spaces, or just $pmbeta_i$ for some $i$?
$endgroup$
– David Hill
Mar 22 at 15:50
1
$begingroup$
Ah, yes thank you, that was a typo! It should have said $ucong mathfrak{sl}_2(mathbb{C})^{oplus l}$.
$endgroup$
– itinerantleopard
Mar 22 at 19:41
add a comment |
$begingroup$
I'm trying to understand the proof of Proposition 7 part (iii) from this paper of Dadok https://www.ams.org/journals/tran/1985-288-01/S0002-9947-1985-0773051-1/S0002-9947-1985-0773051-1.pdf (Statement of the proposition on page 128, proof on page 131), but getting stuck on one line in the the middle.
The set-up is as follows: We have an irreducible self-dual (complex) representation $pi_{lambda}:mathfrak{g}to mathfrak{gl}(V)$ of highest weight $lambda$ (say $v_{lambda}in V$ is a heighest weight vector). There is a distinguished set ${beta_1,dots, beta_l}$ of strongly orthogonal positive roots of $mathfrak{g}$ (their specific definition isn't important for this part), and we are assuming that $lambda$ is in the (real) span of the $beta_i$. We are considering the subalgebra $mathfrak{u}subseteq mathfrak{g}$ generated by the rootspaces of the $pm beta_i$ (note $mathfrak{u}cong mathfrak{sl}_2(mathbb{C})^{oplus l}$).
Self-duality of $pi_{lambda}$ implies that there is a non-degenerate invariant bilinear form $B:Vtimes Vto mathbb{C}$, and to determine whether $pi_{lambda}$ is orthogonal or symplectic we need to figure out whether $B$ is symmetric or skew-symmetric. Now he defines $U_{lambda}subseteq V_{lambda}$ as the $U(mathfrak{u})$-submodule generated by the highest weight vector $v_{lambda}$ (the module $U_{lambda}$ will be simple by the theorem of highest weight, using the fact that $lambda$ is in the span of the $beta_i$). The idea of the argument is that he wants to show that one decide whether $B$ is symmetric or skew-symmetric by looking at the restriction $B:U_{lambda}times U_{lambda}to mathbb{C}$. To that end, he writes:
"If $B:Vto Vto mathbb{C}$ is the nondegenerate bilinear form invariant under $pi_{lambda}$, we see that $B$ must remain non-degenerate on $U_{lambda}times U_{lambda}tomathbb{C}$ since the $mathfrak{u}$-module $U_{lambda}$ appears in $V_{lambda}$ with multiplicity one."
Can anyone help me make sense of this line? I can't even figure out what he means by multiplicity in this context.
Thanks!
abstract-algebra modules representation-theory lie-algebras
asked Mar 21 at 14:53
itinerantleoparditinerantleopard
1379
$endgroup$
I'm trying to understand the proof of Proposition 7 part (iii) from this paper of Dadok https://www.ams.org/journals/tran/1985-288-01/S0002-9947-1985-0773051-1/S0002-9947-1985-0773051-1.pdf (Statement of the proposition on page 128, proof on page 131), but getting stuck on one line in the the middle.
The set-up is as follows: We have an irreducible self-dual (complex) representation $pi_{lambda}:mathfrak{g}to mathfrak{gl}(V)$ of highest weight $lambda$ (say $v_{lambda}in V$ is a heighest weight vector). There is a distinguished set ${beta_1,dots, beta_l}$ of strongly orthogonal positive roots of $mathfrak{g}$ (their specific definition isn't important for this part), and we are assuming that $lambda$ is in the (real) span of the $beta_i$. We are considering the subalgebra $mathfrak{u}subseteq mathfrak{g}$ generated by the rootspaces of the $pm beta_i$ (note $mathfrak{u}cong mathfrak{sl}_2(mathbb{C})^{oplus l}$).
Self-duality of $pi_{lambda}$ implies that there is a non-degenerate invariant bilinear form $B:Vtimes Vto mathbb{C}$, and to determine whether $pi_{lambda}$ is orthogonal or symplectic we need to figure out whether $B$ is symmetric or skew-symmetric. Now he defines $U_{lambda}subseteq V_{lambda}$ as the $U(mathfrak{u})$-submodule generated by the highest weight vector $v_{lambda}$ (the module $U_{lambda}$ will be simple by the theorem of highest weight, using the fact that $lambda$ is in the span of the $beta_i$). The idea of the argument is that he wants to show that one decide whether $B$ is symmetric or skew-symmetric by looking at the restriction $B:U_{lambda}times U_{lambda}to mathbb{C}$. To that end, he writes:
"If $B:Vto Vto mathbb{C}$ is the nondegenerate bilinear form invariant under $pi_{lambda}$, we see that $B$ must remain non-degenerate on $U_{lambda}times U_{lambda}tomathbb{C}$ since the $mathfrak{u}$-module $U_{lambda}$ appears in $V_{lambda}$ with multiplicity one."
Can anyone help me make sense of this line? I can't even figure out what he means by multiplicity in this context.
Thanks!
abstract-algebra modules representation-theory lie-algebras
abstract-algebra modules representation-theory lie-algebras
asked Mar 21 at 14:53
itinerantleoparditinerantleopard
1379
asked Mar 21 at 14:53
itinerantleoparditinerantleopard
1379
edited Mar 22 at 19:42
itinerantleopard
asked Mar 21 at 14:53
itinerantleoparditinerantleopard
1379
asked Mar 21 at 14:53
itinerantleoparditinerantleopard
1379
asked Mar 21 at 14:53
itinerantleoparditinerantleopard
1379
1379
$begingroup$
Why is $mathfrak{u}congmathfrak{sl}_2$? Isn't it generated by multiple root spaces, or just $pmbeta_i$ for some $i$?
$endgroup$
– David Hill
Mar 22 at 15:50
1
$begingroup$
Ah, yes thank you, that was a typo! It should have said $ucong mathfrak{sl}_2(mathbb{C})^{oplus l}$.
$endgroup$
– itinerantleopard
Mar 22 at 19:41
add a comment |
$begingroup$
Why is $mathfrak{u}congmathfrak{sl}_2$? Isn't it generated by multiple root spaces, or just $pmbeta_i$ for some $i$?
$endgroup$
– David Hill
Mar 22 at 15:50
1
$begingroup$
Ah, yes thank you, that was a typo! It should have said $ucong mathfrak{sl}_2(mathbb{C})^{oplus l}$.
$endgroup$
– itinerantleopard
Mar 22 at 19:41
$begingroup$
Why is $mathfrak{u}congmathfrak{sl}_2$? Isn't it generated by multiple root spaces, or just $pmbeta_i$ for some $i$?
$endgroup$
– David Hill
Mar 22 at 15:50
$begingroup$
Why is $mathfrak{u}congmathfrak{sl}_2$? Isn't it generated by multiple root spaces, or just $pmbeta_i$ for some $i$?
$endgroup$
– David Hill
Mar 22 at 15:50
1
1
$begingroup$
Ah, yes thank you, that was a typo! It should have said $ucong mathfrak{sl}_2(mathbb{C})^{oplus l}$.
$endgroup$
– itinerantleopard
Mar 22 at 19:41
$begingroup$
Ah, yes thank you, that was a typo! It should have said $ucong mathfrak{sl}_2(mathbb{C})^{oplus l}$.
$endgroup$
– itinerantleopard
Mar 22 at 19:41
add a comment |
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$begingroup$
Why is $mathfrak{u}congmathfrak{sl}_2$? Isn't it generated by multiple root spaces, or just $pmbeta_i$ for some $i$?
$endgroup$
– David Hill
Mar 22 at 15:50
1
$begingroup$
Ah, yes thank you, that was a typo! It should have said $ucong mathfrak{sl}_2(mathbb{C})^{oplus l}$.
$endgroup$
– itinerantleopard
Mar 22 at 19:41