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non-isotrivial -> non-trivial irreducible
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LSpice
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The question is worded in a way that seems to imply we might speak of other mathematician's tricks, but I'm not sure I know the tricks of even my closest collaborators, except by osmosis; so I hope it's OK if I specify my own "one weird trick". The entirety of my research centres around the idea that, if $\chi$ is a non-trivial character of a compact group $K$ (understood either in the sense of "homomorphism to $\mathbb C^\times$", or the more general sense of $k \mapsto \operatorname{tr} \pi(k)$ for a finite-dimensional, non-isotrivialtrivial, irreducible representation $\pi$ of $K$), then $\int_K \chi(k)\mathrm dk$ equals $0$.

It's amazing the mileage you can get out of this; it usually arises for me when combining Frobenius formula with the first-order approximation in Campbell–Baker–Hausdorff. Combining it with the second-order approximation in CBH gives exponential sums, which in my field we call Gauss sums although that seems to intersect only loosely with how number theorists think of the matter. Curiously, I have never found an application for the third-order approximation.

The question is worded in a way that seems to imply we might speak of other mathematician's tricks, but I'm not sure I know the tricks of even my closest collaborators, except by osmosis; so I hope it's OK if I specify my own "one weird trick". The entirety of my research centres around the idea that, if $\chi$ is a non-trivial character of a compact group $K$ (understood either in the sense of "homomorphism to $\mathbb C^\times$", or the more general sense of $k \mapsto \operatorname{tr} \pi(k)$ for a finite-dimensional, non-isotrivial representation $\pi$ of $K$), then $\int_K \chi(k)\mathrm dk$ equals $0$.

It's amazing the mileage you can get out of this; it usually arises for me when combining Frobenius formula with the first-order approximation in Campbell–Baker–Hausdorff. Combining it with the second-order approximation in CBH gives exponential sums, which in my field we call Gauss sums although that seems to intersect only loosely with how number theorists think of the matter. Curiously, I have never found an application for the third-order approximation.

The question is worded in a way that seems to imply we might speak of other mathematician's tricks, but I'm not sure I know the tricks of even my closest collaborators, except by osmosis; so I hope it's OK if I specify my own "one weird trick". The entirety of my research centres around the idea that, if $\chi$ is a non-trivial character of a compact group $K$ (understood either in the sense of "homomorphism to $\mathbb C^\times$", or the more general sense of $k \mapsto \operatorname{tr} \pi(k)$ for a non-trivial, irreducible representation $\pi$ of $K$), then $\int_K \chi(k)\mathrm dk$ equals $0$.

It's amazing the mileage you can get out of this; it usually arises for me when combining Frobenius formula with the first-order approximation in Campbell–Baker–Hausdorff. Combining it with the second-order approximation in CBH gives exponential sums, which in my field we call Gauss sums although that seems to intersect only loosely with how number theorists think of the matter. Curiously, I have never found an application for the third-order approximation.

Source Link
LSpice
LSpice
  • 12.9k
  • 4
  • 45
  • 69

The question is worded in a way that seems to imply we might speak of other mathematician's tricks, but I'm not sure I know the tricks of even my closest collaborators, except by osmosis; so I hope it's OK if I specify my own "one weird trick". The entirety of my research centres around the idea that, if $\chi$ is a non-trivial character of a compact group $K$ (understood either in the sense of "homomorphism to $\mathbb C^\times$", or the more general sense of $k \mapsto \operatorname{tr} \pi(k)$ for a finite-dimensional, non-isotrivial representation $\pi$ of $K$), then $\int_K \chi(k)\mathrm dk$ equals $0$.

It's amazing the mileage you can get out of this; it usually arises for me when combining Frobenius formula with the first-order approximation in Campbell–Baker–Hausdorff. Combining it with the second-order approximation in CBH gives exponential sums, which in my field we call Gauss sums although that seems to intersect only loosely with how number theorists think of the matter. Curiously, I have never found an application for the third-order approximation.

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