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co.combinatorics – Skew persona with hooks retort

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co.combinatorics – Skew persona with hooks

I maintain requested this in MSE 8 days in the past, aircraft supplied a bounty, and received nothing, so will strive right here.

I might affection to win the worth of the skew characters of the symmetric group, $chi_{lambda/mu}$ within the specific illustration when each $lambda$ and $mu$ are hooks, i.e. $lambda=(a,1^{n-a})$ and $mu=(b,1^{m-b})$ with $n>m$ and $a>b$.

As far as I can graze, there would breathe two approaches to their calculation, however I’m a newbie and I cant labor any of them via. I maintain regarded round, however to no avail.

1) First, the Murnaghan–Nakayama rule says $$chi_{lambda/mu}(nu)=sum_T (-1)^{{rm ht}(T)},$$ the place the sum is taken “over all play-divest tableaux of configuration $lambda/mu$ and kind $nu$”, based on Wikipedia. I assassinate not likely win these tableaux. I denote, I arbitrator $lambda/mu$ isn’t a toy divest if each are hooks; can I quiet sum over toy strips of format $lambda/mu$?

2) I might too write $$chi_{lambda/mu}(nu)=sum_rho c^lambda_{murho}chi_rho(nu),$$ the place $c^lambda_{murho}$ are the Littlewood-Richardson coefficients. I regarded round to graze if they’re recognized when $lambda$ and $mu$ are each hooks, however create nothing.

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