Ben Brubaker, Daniel Bump, and Solomon Friedberg
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.003.0012
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter introduces the Knowability Lemma, which explains when products of Gauss sums associated to elements of a preaccordion are explicitly evaluable as polynomials in q, the order of the ...
More
This chapter introduces the Knowability Lemma, which explains when products of Gauss sums associated to elements of a preaccordion are explicitly evaluable as polynomials in q, the order of the residue class field. It considers an episode in the cartoon associated to the short Gelfand-Tsetlin pattern and the three cases that apply according to the Knowability Lemma, two of which are maximality and knowability. Knowability is not important for the proof that Statement C implies Statement B. The chapter discusses the cases where ε is Class II or Class I, leaving the remaining two cases to the reader. It also describes the variant of the argument for the case that ε is of Class I, again leaving the two other cases to the reader.Less
This chapter introduces the Knowability Lemma, which explains when products of Gauss sums associated to elements of a preaccordion are explicitly evaluable as polynomials in q, the order of the residue class field. It considers an episode in the cartoon associated to the short Gelfand-Tsetlin pattern and the three cases that apply according to the Knowability Lemma, two of which are maximality and knowability. Knowability is not important for the proof that Statement C implies Statement B. The chapter discusses the cases where ε is Class II or Class I, leaving the remaining two cases to the reader. It also describes the variant of the argument for the case that ε is of Class I, again leaving the two other cases to the reader.
Ben Brubaker, Daniel Bump, and Solomon Friedberg
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.003.0009
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter introduces the “Snake Lemma” and uses it to prove the statement that GΓ(t) = GΔ(t′) is “often” true. It first describes an indexing of the Γ preaccordion and of the Δ′ preaccordion; ...
More
This chapter introduces the “Snake Lemma” and uses it to prove the statement that GΓ(t) = GΔ(t′) is “often” true. It first describes an indexing of the Γ preaccordion and of the Δ′ preaccordion; each indexing visits the episodes of the sequence in order from left to right, and after it is finished with an episode, it moves on to the next with no skipping around. Thus both indexings must be in the same episode. If there are resonances, there will be more than one possible pair of snakes. These are obtained through a process of specialization described in the proof. Any one of these pairs of snakes are considered canonical.Less
This chapter introduces the “Snake Lemma” and uses it to prove the statement that GΓ(t) = GΔ(t′) is “often” true. It first describes an indexing of the Γ preaccordion and of the Δ′ preaccordion; each indexing visits the episodes of the sequence in order from left to right, and after it is finished with an episode, it moves on to the next with no skipping around. Thus both indexings must be in the same episode. If there are resonances, there will be more than one possible pair of snakes. These are obtained through a process of specialization described in the proof. Any one of these pairs of snakes are considered canonical.
