Let's break this down and find the value of $X \odot (5,6,7,8)$, where $X = 13124 - 11243 \in R[S_4]$, and we are using the action of the symmetric group $S_4$ on $M = R^4$ as defined.

Step 1: Understand the problem and notation

• $R[S_4]$ is the group algebra of the symmetric group $S_4$, which consists of all permutations on four elements.
• $X = 13124 - 11243$ is an element of $R[S_4]$, and the permutations $13124$ and $11243$ are written in one-line notation.
  • $13124$ likely represents the permutation that sends $(1 \to 1, 2 \to 3, 3 \to 1, 4 \to 2)$.
  • $11243$ represents the permutation sending $(1 \to 1, 2 \to 1, 3 \to 2, 4 \to 3)$.

We are tasked with computing the action of $X = 13124 - 11243$ on the tuple $(5, 6, 7, 8)$.

Step 2: Apply the permutations

Action of $13124$ on $(5, 6, 7, 8)$

The permutation $13124$ acts as follows:

• $1 \to 1$ means the first element remains the same, so the first element of $(5, 6, 7, 8)$ stays 5.
• $2 \to 3$ means the second element becomes the third, so the second element becomes 7.
• $3 \to 1$ means the third element becomes the first, so the third element becomes 5.
• $4 \to 2$ means the fourth element becomes the second, so the fourth element becomes 6.

Thus, $13124$ acting on $(5, 6, 7, 8)$ gives: $13124 \odot (5, 6, 7, 8) = (5, 7, 5, 6)$

Action of $11243$ on $(5, 6, 7, 8)$

The permutation $11243$ acts as follows:

• $1 \to 1$ means the first element remains the same, so the first element stays 5.
• $2 \to 1$ means the second element becomes the first, so the second element becomes 5.
• $3 \to 2$ means the third element becomes the second, so the third element becomes 6.
• $4 \to 3$ means the fourth element becomes the third, so the fourth element becomes 7.

Thus, $11243$ acting on $(5, 6, 7, 8)$ gives: $11243 \odot (5, 6, 7, 8) = (5, 5, 6, 7)$

Step 3: Compute $X \odot (5, 6, 7, 8)$

Now, we can subtract the two results since $X = 13124 - 11243$: $X \odot (5, 6, 7, 8) = (5, 7, 5, 6) - (5, 5, 6, 7)$

Subtracting element-wise: $X \odot (5, 6, 7, 8) = (5 - 5, 7 - 5, 5 - 6, 6 - 7) = (0, 2, -1, -1)$

Final Answer:

The value of $X \odot (5, 6, 7, 8)$ is: $(0, 2, -1, -1)$

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Would you like more details or further clarification? Here are some questions that might help expand your understanding:

1. What is the significance of the group algebra $R[S_4]$ in this context?
2. How is the action of a permutation on a tuple formally defined in terms of group theory?
3. Could we generalize this result for larger symmetric groups $S_n$?
4. What are the properties of $S_4$ that make this specific calculation interesting?
5. How does the group action of symmetric groups apply in fields like representation theory?

Tip: When dealing with permutations, writing out the explicit mappings from positions to values can greatly simplify the process of understanding group actions.