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Let $n_\lambda^K$ be the number all semi-standard Young tableaux of size $K$ with Ferrers diagrams diagram $\lambda$ (i.e. the number of all fillings of $\lambda$ with natural numbers with weakly increasing rows and strictly increasing columns, the content of which sum to $K$).

To each diagram $\lambda$ there corresponds a representation of $GL(2m,C)$ and the sum $\sum_{\mu:l(\mu)\le 2m}N_{\lambda\mu} \mu$ of diagrams $\mu$, into which this representation decomposes upon restriction to $O(2m,C)$ ([1]eq. 25.37).
Here $l(\mu)=\mu_1^\prime + \mu_2^\prime$ denotes the sum of the lengths of the first and the second column of $\mu$.

The multiplicites $N_{\lambda\mu} = \sum_\delta N_{\delta \mu \lambda}$ are the sums of the Littlewood Richardson coefficients $N_{\delta \mu \lambda}$ over all even subdiagrams $\delta$ of $\lambda$, i.e. $\delta_i$ even, $\delta_i\le \lambda_i$, $\delta_1\ge \delta_2\ge \dots$.

Then it is known from string theory, that for $m=12$ and for each $K\ge 2$ the sum of diagrams $\sum_{\lambda\mu:l(\mu) \le 2m} n_\lambda^K N_{\lambda\mu} \mu $ combines with multiplicities $l_\sigma^{K,m}\ge 0$ to families $\sigma$ of diagrams which arise upon restriction of representations $\sigma$ of $O(2m+1,C)$ to $O(2m,C)$ ([1]eq. 25.34)

$\sum_{\lambda\mu:l(\mu)\le 2m} n_\lambda^K N_{\lambda\mu} \mu =
\sum_\sigma l_\sigma^{K,m}(\sum_{\bar\sigma in \sigma}\bar\sigma)$

where $\bar \sigma$ is in the family $\sigma$ if

$\sigma_1\ge \bar\sigma_1\ge\sigma_2\ge\bar\sigma_2\ge \dots $

Does this relation hold for other $m$? Can one show that its validity for $m$ implies it for smaller $m^\prime$?

[1] William Fulton and Joe Harris, Representation Theory, Springer Verlag, New York, 1991

Let $n_\lambda^K$ be the number all semi-standard Young tableaux of size $K$ with Ferrers diagrams diagram $\lambda$ (i.e. the number of all fillings of $\lambda$ with natural numbers with weakly increasing rows and strictly increasing columns, the content of which sum to $K$).

To each diagram $\lambda$ there corresponds a representation of $GL(2m,C)$ and the sum $\sum_{\mu:l(\mu)\le 2m}N_{\lambda\mu} \mu$ of diagrams $\mu$, into which this representation decomposes upon restriction to $O(2m,C)$ ([1]eq. 25.37).
Here $l(\mu)=\mu_1^\prime + \mu_2^\prime$ denotes the sum of the lengths of the first and the second column of $\mu$.

The multiplicites $N_{\lambda\mu} = \sum_\delta N_{\delta \mu \lambda}$ are the sums of the Littlewood Richardson coefficients $N_{\delta \mu \lambda}$ over all even subdiagrams $\delta$ of $\lambda$, i.e. $\delta_i$ even, $\delta_i\le \lambda_i$, $\delta_1\ge \delta_2\ge \dots$.

Then it is known from string theory, that for $m=12$ and for each $K\ge 2$ the sum of diagrams $\sum_{\lambda\mu:l(\mu) \le 2m} n_\lambda^K N_{\lambda\mu} \mu $ combines with multiplicities $l_\sigma^{K,m}\ge 0$ to families $\sigma$ of diagrams which arise upon restriction of representations $\sigma$ of $O(2m+1,C)$ to $O(2m,C)$ ([1]eq. 25.34)

$$\sum_{\lambda\mu:l(\mu)\le 2m} n_\lambda^K N_{\lambda\mu} \mu = \sum_\sigma l_\sigma^{K,m}(\sum_{\bar\sigma in \sigma}\bar\sigma)$$

where $\bar \sigma$ is in the family $\sigma$ if

$$\sigma_1\ge \bar\sigma_1\ge\sigma_2\ge\bar\sigma_2\ge \dots $$

Does this relation hold for other $m$? Can one show that its validity for $m$ implies it for smaller $m^\prime$?

[1] William Fulton and Joe Harris, Representation Theory, Springer Verlag, New York, 1991

Let $n_\lambda^K$ be the number all semi-standard Young tableaux of size $K$ with Ferrers diagrams diagram $\lambda$ (i.e. the number of all fillings of $\lambda$ with natural numbers with weakly increasing rows and strictly increasing columns, the content of which sum to $K$).

To each diagram $\lambda$ there corresponds a representation of $GL(2m,C)$ ($m$ sufficiently large)
and the sum $\sum_{\mu}N_{\lambda\mu} \mu$ of diagrams $\mu$, into which this representation decomposes upon restriction to $O(2m,C)$ ([1]eq. 25.37).

The multiplicites $N_{\lambda\mu} = \sum_\delta N_{\delta \mu \lambda}$ are the sums of the Littlewood Richardson coefficients $N_{\delta \mu \lambda}$ over all even subdiagrams $\delta$ of $\lambda$, i.e. $\delta_i$ even,  $\delta_i\le \lambda_i$, $\delta_1\ge \delta_2\ge \dots$.

The conjecture states:

For each $K\ge 2$ the sum of diagrams $\sum_{\lambda\mu} n_\lambda^K N_{\lambda\mu} \mu $ combine with multiplicities $l_\sigma^K\ge 0$ to families $\sigma$ of diagrams which arise upon restriction of representations $\sigma$ of $O(2m+1,C)$ to $O(2m,C)$ ([1]eq. 25.34)

$\sum_{\lambda\mu} n_\lambda^K N_{\lambda\mu} \mu = \sum_\sigma l_\sigma^K(\sum_{\bar\sigma}\bar\sigma)$

where for each $\sigma$ the family $(\sum_{\bar\sigma}\bar\sigma)$ consists of all diagrams $\bar\sigma$ with

$\sigma_1\ge \bar\sigma_1\ge\sigma_2\ge\bar\sigma_2\ge \dots $

Comment:

It is known, that at each level $K\ge 2$ the exited string states with vanishing momentum span a vector space which allows a representation of $O(25)$. This proves the conjecture to hold for all diagrams $\mu$ where the sum of the lengths $\mu^\prime_1$ and $\mu^\prime_2$ of the first and second column does not exceed $24$.

If the conjecture can be shown to hold without the restriction $\mu^\prime_1 + \mu^\prime_2\le 24$ then the bosonic string is consistent in arbitrary dimension and the dimension 26 is just an artefact of an unsuitable construction of the generators of Lorentz transformations in the lightcone formulation.

[1] William Fulton and Joe Harris, Representation Theory, Springer Verlag, New York, 1991

Let $n_\lambda^K$ be the number all semi-standard Young tableaux of size $K$ with Ferrers diagrams diagram $\lambda$ (i.e. the number of all fillings of $\lambda$ with natural numbers with weakly increasing rows and strictly increasing columns, the content of which sum to $K$).

To each diagram $\lambda$ there corresponds a representation of $GL(2m,C)$ and the sum $\sum_{\mu:l(\mu)\le 2m}N_{\lambda\mu} \mu$ of diagrams $\mu$, into which this representation decomposes upon restriction to $O(2m,C)$ ([1]eq. 25.37).
Here $l(\mu)=\mu_1^\prime + \mu_2^\prime$ denotes the sum of the lengths of the first and the second column of $\mu$.

The multiplicites $N_{\lambda\mu} = \sum_\delta N_{\delta \mu \lambda}$ are the sums of the Littlewood Richardson coefficients $N_{\delta \mu \lambda}$ over all even subdiagrams $\delta$ of $\lambda$, i.e. $\delta_i$ even,  $\delta_i\le \lambda_i$, $\delta_1\ge \delta_2\ge \dots$.

Then it is known from string theory, that for $m=12$ and for each $K\ge 2$ the sum of diagrams $\sum_{\lambda\mu:l(\mu) \le 2m} n_\lambda^K N_{\lambda\mu} \mu $ combines with multiplicities $l_\sigma^{K,m}\ge 0$ to families $\sigma$ of diagrams which arise upon restriction of representations $\sigma$ of $O(2m+1,C)$ to $O(2m,C)$ ([1]eq. 25.34)

$\sum_{\lambda\mu:l(\mu)\le 2m} n_\lambda^K N_{\lambda\mu} \mu =
\sum_\sigma l_\sigma^{K,m}(\sum_{\bar\sigma in \sigma}\bar\sigma)$

where $\bar \sigma$ is in the family $\sigma$ if

$\sigma_1\ge \bar\sigma_1\ge\sigma_2\ge\bar\sigma_2\ge \dots $

Does this relation hold for other $m$? Can one show that its validity for $m$ implies it for smaller $m^\prime$?

[1] William Fulton and Joe Harris, Representation Theory, Springer Verlag, New York, 1991

Let $n_\lambda^K$ be the number all semi-standard Young tableaux of size $K$ with Ferrers diagrams diagram $\lambda$ (i.e. the number of all fillings of $\lambda$ with natural numbers with weakly increasing rows and strictly increasing columns, the content of which sum to $K$).

To each diagram $\lambda$ there corresponds a representation of $GL(2m,C)$ ($m$ sufficiently large)
and the sum $\sum_{\mu}N_{\lambda\mu} \mu$ of diagrams $\mu$, into which this representation decomposes upon restriction to $O(2m,C)$ ([1]eq. 25.37).

The multiplicites $N_{\lambda\mu} = \sum_\delta N_{\delta \mu \lambda}$ are the sums of the Littlewood Richardson coefficients $N_{\delta \mu \lambda}$ over all even subdiagrams $\delta$ of $\lambda$, i.e. $\delta_i$ even, $\delta_i\le \lambda_i$, $\delta_1\ge \delta_2\ge \dots$.

The conjecture states:

For each $K\ge 2$ the sum of diagrams $\sum_{\lambda\mu} n_\lambda^K N_{\lambda\mu} \mu $ combine with multiplicities $l_\sigma^K\ge 0$ to families $\sigma$ of diagrams which arise upon restriction of representations $\sigma$ of $O(2m+1,C)$ to $O(2m,C)$ ([1]eq. 25.34)

$\sum_{\lambda\mu} n_\lambda^K N_{\lambda\mu} \mu =
\sum_\sigma l_\sigma^K(\sum_{\bar\sigma}\bar\sigma)$

where for each $\sigma$ the family $(\sum_{\bar\sigma}\bar\sigma)$ consists of all diagrams $\bar\sigma$ with

$\sigma_1\ge \bar\sigma_1\ge\sigma_2\ge\bar\sigma_2\ge \dots $

Comment:

It is known, that at each level $K\ge 2$ the exited string states with vanishing momentum span a vector space which allows a representation of $O(25)$. This proves the conjecture to hold for all diagrams $\mu$ where the sum of the lengths $\mu^\prime_1$ and $\mu^\prime_2$ of the first and second column does not exceed $24$.

If the conjecture can be shown to hold without the restriction $\mu^\prime_1 + \mu^\prime_2\le 24$ then the bosonic string is consistent in arbitrary dimension and the dimension 26 is just an artefact of an unsuitable construction of the generators of Lorentz transformations in the lightcone formulation.

[1] William Fulton and Joe Harris, Representation Theory, Springer Verlag, New York, 1991

Let $n_\lambda^K$ be the number all semi-standard Young tableaux of size $K$ with Ferrers diagrams diagram $\lambda$ (i.e. the number of all fillings of $\lambda$ with natural numbers with weakly increasing rows and strictly increasing columns, the content of which sum to $K$).

To each diagram $\lambda$ there corresponds a representation of $GL(2m,C)$ ($m$ sufficiently large)
and the sum $\sum_{\mu}N_{\lambda\mu} \mu$ of diagrams $\mu$, into which this representation decomposes upon restriction to $O(2m,C)$ ([1]eq. 25.37).

The multiplicites $N_{\lambda\mu} = \sum_\delta N_{\delta \mu \lambda}$ are the sums of the Littlewood Richardson coefficients $N_{\delta \mu \lambda}$ over all even subdiagrams $\delta$ of $\lambda$, i.e. $\delta_i$ even, $\delta_i\le \lambda_i$, $\delta_1\ge \delta_2\ge \dots$.

The conjecture states:

For each $K\ge 2$ the sum of diagrams $\sum_{\lambda\mu} n_\lambda^K N_{\lambda\mu} \mu $ combine with multiplicities $l_\sigma^K\ge 0$ to families $\sigma$ of diagrams which arise upon restriction of representations $\sigma$ of $O(2m+1,C)$ to $O(2m,C)$ ([1]eq. 25.34)

$\sum_{\lambda\mu} n_\lambda^K N_{\lambda\mu} \mu = \sum_\sigma l_\sigma^K(\sum_{\bar\sigma}\bar\sigma)$

where for each $\sigma$ the family $(\sum_{\bar\sigma}\bar\sigma)$ consists of all diagrams $\bar\sigma$ with

$\sigma_1\ge \bar\sigma_1\ge\sigma_2\ge\bar\sigma_2\ge \dots $

Comment:

It is known, that at each level $K\ge 2$ the exited string states with vanishing momentum span a vector space which allows a representation of $O(25)$. This proves the conjecture to hold for all diagrams $\mu$ where the sum of the lengths $\mu^\prime_1$ and $\mu^\prime_2$ of the first and second column does not exceed $24$.

If the conjecture can be shown to hold without the restriction $\mu^\prime_1 + \mu^\prime_2\le 24$ then the bosonic string is consistent in arbitrary dimension and the dimension 26 is just an artefact of an unsuitable construction of the generators of Lorentz transformations in the lightcone formulation.

[1] William Fulton and Joe Harris, Representation Theory, Springer Verlag, New York, 1991