# The Jantzen-Schaper theorem

Does anybody have an electronic copy of Schaper's PhD thesis:

K.D. SCHAPER, ‘Charakterformeln fur Weyl-Moduln und Specht-Moduln in Primcharacteristik’, Diplomarbeit, Bonn, 1981.

I would like to understand how to pass between Jantzen's formulation of the "Jantzen sum formula" for ${\rm GL}_n$ in terms of reflections in a type $A$ geometry, to James and Mathas' combinatorial treatment given in terms of hooks in the Young diagram.

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Note that this is not actually a Ph.D. thesis, but a somewhat lower level Diplom thesis. It has had influence in the literature, which is by now extensive. But I'm doubtful that the original document will add much. It's also unlikely to have been scanned for internet access, but of course there are some paper copies. I believe I still have a copy at UMass. But the later literature is much more likely to be helpful: papers by James-Mathas, Fayer, and others. – Jim Humphreys May 15 '13 at 13:52

To expand my comment further, I do in fact have a copy of the 50+ page typewritten double-spaced document by Klaus-Dieter Schaper (dated June 1981). As usual this Diplomarbeit was not published, nor did Schaper himself apparently continue in mathematics. Though I'm not at all up-to-date on the problems surrounding Specht modules and decomposition numbers for symmetric groups, I might be able to answer specific questions about Schaper's actual work.

In any case, Schaper and a few others working on symmetric groups do include references to the influential early paper by Carter and Lusztig in Math. Z. 136 (1974): On the modular representations of the general linear and symmetric groups (available at the German archive GDZ). This paper preceded most of Jantzen's work, but it shows in a concrete way how to pass from the tensor representations of the general linear group (expressed in highest weight language) to the representations of symmetric groups on zero weight spaces. This connection is classical in characteristic 0, but leads to numerous complications in characteristic $p$ when this prime divides the order of a symmetric group acting on a tensor power of the natural module.

Jantzen showed how to start with a classical characteristic 0 module for the general (or special) linear group and then work with an explicit filtration of the resulting characteristic $p$ Weyl module, using his filtration coming from a contravariant form, etc. The resulting sum formula expresses formal characters using the dot-action of an associated (Langlands dual) affine Weyl group on the highest weight. In turn, by realizing an irreducible representation of a Weyl group on the zero weight space in characteristic 0, one gets by reduction mod $p$ a Jantzen-type filtrration in the resulting "Specht module" studied earlier by Gordon James. Then a sum formula, etc.

What you might find in Carter-Lusztig's explicit treatment in 4.2 is a transition from the affine Weyl group action to the symmetric group picture involving Young diagrams and raising squares. While the general theory has had a lot of further development, the ideas in Schaper's work (heavily influenced by Jantzen, of course) owe much to this earlier framework.

Matt Fayers might have more precise things to add. (Sorry for my earlier misspelling of his name, which happens to me all the time.)

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Thanks Jim! I'll have a look at that. I was looking at $V(4,1)$ for $SL_3$ in characteristic 3. I know that I should be able to rewrite the sum $\sum_{i>0}V^i(4,1)$ as V(3,0)+V(0,3)+V(0,0) But I couldn't get from Delta(4,1) to Delta(0,0) using a single reflection in the geometry, I had to take 3 reflections and compose. Is my problem that I'm focussing on SL_3, rather than GL_3? – Chris Bowman May 16 '13 at 1:37
Sorry $\Delta=V$, I swapped notation half-way through – Chris Bowman May 16 '13 at 1:37
@Chris: $\mathrm{SL}_3$ is standard here. Jantzen's method is subtle: you may get a non-dominant weight by reflecting across a (-$\rho$-shifted) affine hyperplane and must then interpret his $\chi(\mu)$ as $\pm$ linked Weyl character. The sum formula gives formally three terms $\chi(\mu)$, two from reflections relative to highest root with one indirectly linked to the 0 weight (with sign = 1). Draw alcove picture! (Besides RAGS, see his short survey, pp. 291-300 in the 1978 Durham proceedings Finite Simple Groups II, Academic Press, 1980.) – Jim Humphreys May 16 '13 at 19:30
P.S. Jantzen's sum formula is computable but doesn't usually pin down the precise filtration layers. Here it's clear: your Weyl module $V(4,1)$ has Weyl dim 35, with simple quotient on layer 0, of dim 21 (from restricted weights and Steinberg tensor product); layer 1 then has three simples of dim 1, 3, 3, while layer 2 has one simple of dim 7. (Here the five composition factors with multiplicity 1 can most easily be seen from Jantzen's generic decomposition formula, which degenerates preictably near a Weyl chamber wall.) – Jim Humphreys May 16 '13 at 20:27
So, is there some obvious way of passing from non-dominant weights to dominant weights? Is this similar to the cancellation one sees close to the walls when using the generic patterns for Weyl modules? For example, in the SL_3 case I think these are mentioned in papers of Doty (and perhaps Sullivan). Would it be fair to say that James and Mathas' formulation is easier to work with (for a given example) in the GL_n case, than Jantzen's own formulation? – Chris Bowman May 17 '13 at 9:17

I just came across this. If you still want a copy of Schaper's thesis let me know as I have a scanned copy (your Jussieu email address no longer works and I can't find a new one).

What Schaper did really amounted to a combinatorial translation of Jantzen's sum formula in terms of the dot action into combinatorics. I think that Schaper's thesis was largely motivated by a (flawed) preprint of Dave Benson's which claimed to prove the sum formula for the symmetric groups. In turn Benson was motivated by James and Murphy's formula for the Gram determinants of the Specht modules. Schaper's thesis is a nice read and for me it was only by reading his thesis that I realised that in order to prove the sum formula "all" that we needed to do was to compute the determinants of all the weight spaces of the Weyl modules. This led directly to my first paper with Gordon on sum formulas.

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