For natural numbers $a,b$ with $b\leq n-1$, let $V_{ (a|b)}$ be the irreducible representation of $GL_n$ with highest weight vector $(a+1, 1^b, 0^{n-b-1})$ where the exponentiation denotes repetition.

The Giambelli identity states that if $a_1 > \dots > a_r$ and $b_1 > \dots > b_r$ are natural numbers with $b_1 \leq n-1$ then the determinant in the representation ring of the matrix $M$ with entries $M_{ij} = V_{ (a_i |b_j)} $ is an irreducible representation of $GL_n$ with highest weight vector $$( a_1 + 1,\dots, a_r +r, r^{b_r}, (r-1)^{b_{r-1} -b_r-1}, \dots, 1^{ b_1 - b_{2}-1} , 0 ^{n-1-b_1}).$$

Technically, the Giambelli identity is an identity in the ring of symmetric functions in infinitely many variables. We deduce this using the fact that the representation ring of $GL_n$ is the quotient ring of symmetric functions in $n$ variables, where each Schur function is sent to an irreducible representation or zero depending on the number of parts.

Let $a_1 > \dots a_r, b_1> \dots > b_{2r}, c_1> \dots c_r$ be natural numbers with $b_1 \leq n-1$.

Let $M$ be the matrix whose entries are $M_{ij} = V_{(a_i|b_j)}$ if $1 \leq i \leq r$ and $M_{ij} = V_{( c_{2r-i} | n-1-b_j )}^\vee$ if $ r+1 \leq i \leq 2r$.

Is $\det M$ an irreducible representation? Is it the one with highest weight vector $$(a_1+1,\dots, a+r+r, r^{b_{2r} }, (r-1)^{b_{2r-1}-b_{2r} -1},\dots, (-r+1)^{ b_1-b_2 -1}, (-r)^{n-1-b_{1}}, -c_r-r, \dots, -c_1-1)?$$

If not, does some similar formula hold for the class of this irreducible representation in the representation ring?

The motivation is that this would help generalize my work in arXiv:1810.01303 on the CFKRS conjecture for integral moments, from the moments conjectures to the ratios conjecture.

It doesn't seem possible to deduce this from any nice identity purely in the ring of symmetric functions, so the same approach might not work here.

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    $\begingroup$ Theorem 3.5 here seems to be the relevant identity arxiv.org/abs/math/0504487 $\endgroup$ Nov 24 '18 at 23:51
  • $\begingroup$ @GjergjiZaimi I can confirm that this is exactly what I need, want to post it as an answer? $\endgroup$
    – Will Sawin
    Nov 25 '18 at 2:30

Yes, your prediction is correct. The determinant identity in this case is theorem 3.5 in Division and the Giambelli Identity, by Wu and Yang (also published at Linear Algebra Appl. 406 (2005), 301-309).


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