Notation for (skew) Young Tableaux in Hubsch
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I'm looking at the notation given for Young tableau in Hubsch's book "Calabi-Yau Manifolds: A Bestiary for Physicists" - see Chapter 3, p. 96.
A Young tableau (for a $U(n)$ representation) is denoted by
$(b_1, dots, b_n) , , quad b_r leq b_{r+1} $
and the $b_{r}$ denote the number of boxes in the $r$-th row of the Young tableau, counting upwards from the bottom row.
This is all very clear until the mention of having negative values of $b_r$ - I've not really come across this before.
Further, the covariant ($v_mu$) and contravariant ($v^mu$) vector representations are denoted by
$(-1, 0, dots , 0)$ and $(0, dots, 0, 1)$
respectively.
The only way this makes sense is if we consider "skew"-tableau, but I'm not too clear about how these map to the usual notion of representations. I'm sure this is well understood somewhere as I've seen this employed in calculations of cohomologies on projective spaces but I can't find a simpler explanation of the notation.
group-theory algebraic-geometry representation-theory projective-geometry string-theory
asked Nov 28 at 16:34
nonreligious
133
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0
down vote
favorite
I'm looking at the notation given for Young tableau in Hubsch's book "Calabi-Yau Manifolds: A Bestiary for Physicists" - see Chapter 3, p. 96.
A Young tableau (for a $U(n)$ representation) is denoted by
$(b_1, dots, b_n) , , quad b_r leq b_{r+1} $
and the $b_{r}$ denote the number of boxes in the $r$-th row of the Young tableau, counting upwards from the bottom row.
This is all very clear until the mention of having negative values of $b_r$ - I've not really come across this before.
Further, the covariant ($v_mu$) and contravariant ($v^mu$) vector representations are denoted by
$(-1, 0, dots , 0)$ and $(0, dots, 0, 1)$
respectively.
The only way this makes sense is if we consider "skew"-tableau, but I'm not too clear about how these map to the usual notion of representations. I'm sure this is well understood somewhere as I've seen this employed in calculations of cohomologies on projective spaces but I can't find a simpler explanation of the notation.
group-theory algebraic-geometry representation-theory projective-geometry string-theory
asked Nov 28 at 16:34
nonreligious
133
FYI: reposting this from the Physics SE.
– nonreligious
Nov 28 at 16:34
There seems to be some garbled terminology here -- the "Young tableaux" you are talking about look like partitions, or the "rational" version of partitions introduced by Stembridge (John R. Stembridge, Rational tableaux and the tensor algebra of $gl_n$, Journal of Combinatorial Theory, Series A 46, pp. 79--120 (1987)).
– darij grinberg
Dec 2 at 10:13
@darijgrinberg Thanks for your comment - I think you're correct. Is there an example of how I can view these in relation to the representations of $U(n)$, or vector bundles on $mathbb{P}^n$ ?
– nonreligious
Dec 3 at 21:35
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm looking at the notation given for Young tableau in Hubsch's book "Calabi-Yau Manifolds: A Bestiary for Physicists" - see Chapter 3, p. 96.
A Young tableau (for a $U(n)$ representation) is denoted by
$(b_1, dots, b_n) , , quad b_r leq b_{r+1} $
and the $b_{r}$ denote the number of boxes in the $r$-th row of the Young tableau, counting upwards from the bottom row.
This is all very clear until the mention of having negative values of $b_r$ - I've not really come across this before.
Further, the covariant ($v_mu$) and contravariant ($v^mu$) vector representations are denoted by
$(-1, 0, dots , 0)$ and $(0, dots, 0, 1)$
respectively.
The only way this makes sense is if we consider "skew"-tableau, but I'm not too clear about how these map to the usual notion of representations. I'm sure this is well understood somewhere as I've seen this employed in calculations of cohomologies on projective spaces but I can't find a simpler explanation of the notation.
group-theory algebraic-geometry representation-theory projective-geometry string-theory
asked Nov 28 at 16:34
nonreligious
133
I'm looking at the notation given for Young tableau in Hubsch's book "Calabi-Yau Manifolds: A Bestiary for Physicists" - see Chapter 3, p. 96.
A Young tableau (for a $U(n)$ representation) is denoted by
$(b_1, dots, b_n) , , quad b_r leq b_{r+1} $
and the $b_{r}$ denote the number of boxes in the $r$-th row of the Young tableau, counting upwards from the bottom row.
This is all very clear until the mention of having negative values of $b_r$ - I've not really come across this before.
Further, the covariant ($v_mu$) and contravariant ($v^mu$) vector representations are denoted by
$(-1, 0, dots , 0)$ and $(0, dots, 0, 1)$
respectively.
The only way this makes sense is if we consider "skew"-tableau, but I'm not too clear about how these map to the usual notion of representations. I'm sure this is well understood somewhere as I've seen this employed in calculations of cohomologies on projective spaces but I can't find a simpler explanation of the notation.
group-theory algebraic-geometry representation-theory projective-geometry string-theory
group-theory algebraic-geometry representation-theory projective-geometry string-theory
asked Nov 28 at 16:34
nonreligious
133
asked Nov 28 at 16:34
nonreligious
133
asked Nov 28 at 16:34
nonreligious
133
asked Nov 28 at 16:34
nonreligious
133
asked Nov 28 at 16:34
nonreligious
133
133
FYI: reposting this from the Physics SE.
– nonreligious
Nov 28 at 16:34
There seems to be some garbled terminology here -- the "Young tableaux" you are talking about look like partitions, or the "rational" version of partitions introduced by Stembridge (John R. Stembridge, Rational tableaux and the tensor algebra of $gl_n$, Journal of Combinatorial Theory, Series A 46, pp. 79--120 (1987)).
– darij grinberg
Dec 2 at 10:13
@darijgrinberg Thanks for your comment - I think you're correct. Is there an example of how I can view these in relation to the representations of $U(n)$, or vector bundles on $mathbb{P}^n$ ?
– nonreligious
Dec 3 at 21:35
add a comment |
FYI: reposting this from the Physics SE.
– nonreligious
Nov 28 at 16:34
There seems to be some garbled terminology here -- the "Young tableaux" you are talking about look like partitions, or the "rational" version of partitions introduced by Stembridge (John R. Stembridge, Rational tableaux and the tensor algebra of $gl_n$, Journal of Combinatorial Theory, Series A 46, pp. 79--120 (1987)).
– darij grinberg
Dec 2 at 10:13
@darijgrinberg Thanks for your comment - I think you're correct. Is there an example of how I can view these in relation to the representations of $U(n)$, or vector bundles on $mathbb{P}^n$ ?
– nonreligious
Dec 3 at 21:35
FYI: reposting this from the Physics SE.
– nonreligious
Nov 28 at 16:34
FYI: reposting this from the Physics SE.
– nonreligious
Nov 28 at 16:34
There seems to be some garbled terminology here -- the "Young tableaux" you are talking about look like partitions, or the "rational" version of partitions introduced by Stembridge (John R. Stembridge, Rational tableaux and the tensor algebra of $gl_n$, Journal of Combinatorial Theory, Series A 46, pp. 79--120 (1987)).
– darij grinberg
Dec 2 at 10:13
There seems to be some garbled terminology here -- the "Young tableaux" you are talking about look like partitions, or the "rational" version of partitions introduced by Stembridge (John R. Stembridge, Rational tableaux and the tensor algebra of $gl_n$, Journal of Combinatorial Theory, Series A 46, pp. 79--120 (1987)).
– darij grinberg
Dec 2 at 10:13
@darijgrinberg Thanks for your comment - I think you're correct. Is there an example of how I can view these in relation to the representations of $U(n)$, or vector bundles on $mathbb{P}^n$ ?
– nonreligious
Dec 3 at 21:35
@darijgrinberg Thanks for your comment - I think you're correct. Is there an example of how I can view these in relation to the representations of $U(n)$, or vector bundles on $mathbb{P}^n$ ?
– nonreligious
Dec 3 at 21:35
add a comment |
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FYI: reposting this from the Physics SE.
– nonreligious
Nov 28 at 16:34
There seems to be some garbled terminology here -- the "Young tableaux" you are talking about look like partitions, or the "rational" version of partitions introduced by Stembridge (John R. Stembridge, Rational tableaux and the tensor algebra of $gl_n$, Journal of Combinatorial Theory, Series A 46, pp. 79--120 (1987)).
– darij grinberg
Dec 2 at 10:13
@darijgrinberg Thanks for your comment - I think you're correct. Is there an example of how I can view these in relation to the representations of $U(n)$, or vector bundles on $mathbb{P}^n$ ?
– nonreligious
Dec 3 at 21:35