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Miguel
Miguel
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Sum of the dimensions of the rational irreducible representations of $S_k \times S_j$

For a group $G$ and a field $K$ let $S(G,K)$ be the sum of the dimensions of the irreducible K representations of $G$. Note that $S(G,\mathbb{C})< |G|$. It's not difficult to prove that if $n \ge 6$ then $S(S_n,\mathbb{C}) < (n-2)!(n-2)-n$. I'm interested in "good" bounds (not necessarily the best but at least significantly better than $|G|$).

I want to know if there is a relation between $S(G,\mathbb{Q})$ and $S(G,\mathbb{C})$. I need this result for the subgroups of $S_n$ isomorphic to $S_j \times S_{n-j}$. Specifically I need a bound for $S(S_k \times S_{n-k}, \mathbb{Q})$ the best as possible. I'm asking a relation with respect to $S(S_k \times S_{n-k}, \mathbb{C})$ because it looks that the problem is easier here since the complex irreps are given by tensor products (not sure if it's true for $\mathbb{Q}$ representations).

Sum of the dimensions of the rational irreducible representations of $S_k \times S_j$

For a group $G$ and a field $K$ let $S(G,K)$ be the sum of the dimensions of the irreducible K representations of $G$. Note that $S(G,\mathbb{C})< |G|$. It's not difficult to prove that if $n \ge 6$ then $S(S_n,\mathbb{C}) < (n-2)!(n-2)-n$. I'm interested in "good" bounds (not necessarily the best but at least significantly better than $|G|$).

I want to know if there is a relation between $S(G,\mathbb{Q})$ and $S(G,\mathbb{C})$. I need this result for the subgroups of $S_n$ isomorphic to $S_j \times S_{n-j}$. Specifically I need a bound for $S(S_k \times S_{n-k}, \mathbb{Q})$ the best as possible. I'm asking a relation with respect to $S(S_k \times S_{n-k}, \mathbb{C})$ because it looks that the problem is easier here since the complex irreps are given by tensor products (not sure if it's true for $\mathbb{Q}$ representations).

representations of $S_k \times S_j$

For a group $G$ and a field $K$ let $S(G,K)$ be the sum of the dimensions of the irreducible K representations of $G$. Note that $S(G,\mathbb{C})< |G|$. It's not difficult to prove that if $n \ge 6$ then $S(S_n,\mathbb{C}) < (n-2)!(n-2)-n$. I'm interested in "good" bounds (not necessarily the best but at least significantly better than $|G|$). I need a bound for $S(S_k \times S_{n-k}, \mathbb{Q})$ the best as possible.

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Miguel
Miguel
  • 163
  • 3

For a group $G$ and a field $K$ let $S(G,K)$ be the sum of the dimensions of the irreducible K representations of $G$. Note that $S(G,\mathbb{C})< |G|$. It's not difficult to prove that if $n \ge 6$ then $S(S_n,\mathbb{C}) < (n-2)!(n-2)-n$. I'm interested in "good" bounds (not necessarily the best but at least significantly better than $|G|$).

I want to know if there is a relation between $S(G,\mathbb{Q})$ and $S(G,\mathbb{C})$. I need this result for the subgroups of $S_n$ isomorphic to $S_j \times S_{n-j}$. Specifically I need a bound for $S(S_k \times S_{n-k}, \mathbb{Q})$ the best as possible. I'm asking a relation with respect to $S(S_k \times S_{n-k}, \mathbb{C})$ because it looks that the problem is easier here since the complex irreps are given by tensor products (not sure if it's true for $\mathbb{Q}$ representations).

For a group $G$ and a field $K$ let $S(G,K)$ be the sum of the dimensions of the irreducible K representations of $G$. Note that $S(G,\mathbb{C})< |G|$. It's not difficult to prove that if $n \ge 6$ then $S(S_n,\mathbb{C}) < (n-2)!(n-2)-n$. I'm interested in "good" bounds (not necessarily the best but at least significantly better than $|G|$).

I want to know if there is a relation between $S(G,\mathbb{Q})$ and $S(G,\mathbb{C})$. I need this result for the subgroups of $S_n$ isomorphic to $S_j \times S_{n-j}$. Specifically I need a bound for $S(S_k \times S_{n-k}, \mathbb{Q})$ the best as possible.

For a group $G$ and a field $K$ let $S(G,K)$ be the sum of the dimensions of the irreducible K representations of $G$. Note that $S(G,\mathbb{C})< |G|$. It's not difficult to prove that if $n \ge 6$ then $S(S_n,\mathbb{C}) < (n-2)!(n-2)-n$. I'm interested in "good" bounds (not necessarily the best but at least significantly better than $|G|$).

I want to know if there is a relation between $S(G,\mathbb{Q})$ and $S(G,\mathbb{C})$. I need this result for the subgroups of $S_n$ isomorphic to $S_j \times S_{n-j}$. Specifically I need a bound for $S(S_k \times S_{n-k}, \mathbb{Q})$ the best as possible. I'm asking a relation with respect to $S(S_k \times S_{n-k}, \mathbb{C})$ because it looks that the problem is easier here since the complex irreps are given by tensor products (not sure if it's true for $\mathbb{Q}$ representations).

Source Link
Miguel
Miguel
  • 163
  • 3

Sum of the dimensions of the rational irreducible representations of $S_k \times S_j$

For a group $G$ and a field $K$ let $S(G,K)$ be the sum of the dimensions of the irreducible K representations of $G$. Note that $S(G,\mathbb{C})< |G|$. It's not difficult to prove that if $n \ge 6$ then $S(S_n,\mathbb{C}) < (n-2)!(n-2)-n$. I'm interested in "good" bounds (not necessarily the best but at least significantly better than $|G|$).

I want to know if there is a relation between $S(G,\mathbb{Q})$ and $S(G,\mathbb{C})$. I need this result for the subgroups of $S_n$ isomorphic to $S_j \times S_{n-j}$. Specifically I need a bound for $S(S_k \times S_{n-k}, \mathbb{Q})$ the best as possible.