Timeline for Symmetric polynoms are Hopf algebra ? What for one needs co-product ?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Sep 13, 2016 at 15:20 | comment | added | darij grinberg | ... called the "alphabet doubling trick", and the difficulties of making it rigorous are a reason why it rarely appears as a definition of $\Delta$ (another is that it is easy to define $\Delta\left(M_\alpha\right)$ directly); but it can be done, and it can be useful. Similar tricks work for some noncommutative generalizations of quasisymmetric functions. So the idea of alphabet doubling is not unique to symmetric functions. | |
| Sep 13, 2016 at 15:19 | comment | added | darij grinberg | Just a belated footnote: Quasisymmetric functions also allow a coproduct defined by the same idea, even though the order does matter. The details are subtler and more complicated; $\Delta\left(s\right)$ is now obtained by applying $f$ to $x_0, x_1, x_2, \ldots, y_0, y_1, y_2, \ldots$ (to use Marc's zero-based indexing), and some work is needed to make sense of this (since the variables are no longer mapped directly to the infinitely many variables $x_0, x_1, x_2, \ldots$; some guests in this not-quite-Hilbert hotel would have to walk an infinite distance to their rooms). This is ... | |
| Sep 13, 2016 at 15:16 | history | edited | darij grinberg | CC BY-SA 3.0 |
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| Jan 19, 2012 at 6:52 | comment | added | Alexander Chervov | Do you mean that setup of "Hilbert hotel" (I am not clearly understanding it) is equivalent to symmetric function setup ? Is it easy to see this (if it is true) or it is non-trivial ? | |
| Jan 19, 2012 at 6:47 | comment | added | Marc van Leeuwen | Well there is the part "order doesn't matter" that depends on having symmetric power series (as I have now emphasised in the answer). In the Hilbert hotel one needs to be able to freely permute guests among rooms in order to operate efficiently. Due to this I do not see how to apply this simple idea in fundamentally different situations. | |
| Jan 19, 2012 at 6:41 | history | edited | Marc van Leeuwen | CC BY-SA 3.0 |
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| Jan 19, 2012 at 4:54 | comment | added | Alexander Chervov | That seems to be great point ! But it seems very general - not much using particular structure of "symmetric functions" (or I am missing something?), are their some other natural examples where one can introduce co-product using the same idea ? | |
| Jan 18, 2012 at 22:51 | history | answered | Marc van Leeuwen | CC BY-SA 3.0 |