Abstract
Fix two finite sets I = { 1, 2, …, n}, J = { 1, 2, …, m} and consider a rectangular table with n columns and m rows numbered by the elements of I and J respectively in such a way that indices I increase horizontally from left to right, and indices j increase vertically from bottom to top.
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Notes
- 1.
Or just an effective word if a is clear from the context or inessential to the discussion.
- 2.
Note that if i = j and both L i, D i act effectively, then L iD i and D iL i move one ball from the (i, i) cell to the (i − 1, i − 1) cell in two different ways.
- 3.
Or dense downward.
- 4.
We follow Sect. 3.4.2 on p. 63 and think of a Young diagram as an infinite sequence of nonincreasing nonnegative integers tending to zero.
- 5.
Recall that this means that every element of I appears in the tableau exactly once; see Sect. 4.2.3 on p. 81.
- 6.
That is, σ ∈ S n satisfying σ 2 = 1.
- 7.
W. Fulton. Young Tableaux with Applications to Representation Theory and Geometry. LMS Student Texts 35, CUP (1997).
- 8.
In combinatorics, the central symmetry a ↦ a ∗ is called the Schützenberger involution; see Problem 4.10 on p. 98.
- 9.
V. I. Danilov, G. A. Koshevoy. “Arrays and the Combinatorics of Young Tableaux.” Russian Math. Surveys 60:2 (2005), 269–334.
- 10.
See Sect. 13.2 of Algebra I.
- 11.
See Sect. 13.1 of Algebra I.
- 12.
Since every Young diagram λ = (λ 1, λ 2, …, λ m) can be viewed as a vector in \(\mathbb{Z}_{\geqslant 0}^{m}\), the domination relation \(\lambda ' \trianglerighteq \lambda ''\) is well defined.
- 13.
Recall that the weight of a Young diagram λ is the total number of cells in λ.
- 14.
See Sect. 3.3 on p. 61.
- 15.
See Sect. 3.2 on p. 60.
- 16.
See Sect. 4.2.4 on p. 82.
- 17.
See Sect. 3.2 on p. 60.
- 18.
See Sect. 3.3 on p. 61.
- 19.
- 20.
See Sect. 10.3.1 of Algebra I.
- 21.
- 22.
See formula (4.16) on p. 91.
- 23.
Recall that the first tableau is the row scan of the D-condensation of the graph of g: I ≃ J, whereas the second is the column scan of the L-condensation of the same graph.
- 24.
That is, a map commuting with all the vertical operations D j, U j.
- 25.
That is, λ ≠ μ and for every η, \(\lambda \trianglerighteq \eta \trianglerighteq \mu\) forces either λ = η or η = μ.
- 26.
Formally, \(\gamma _{i} ={\bigl (\lambda _{i} - i + 1, 1^{\lambda _{i}^{t}-i}\bigr )}\) for every i = 1, 2, … , k.
- 27.
That is, the Young diagram encoding the DL-condensation of the array; see Sect. 4.2.2 on p. 80.
- 28.
See Sect. 1.4 of Algebra I.
- 29.
Note that this forces the shape of P(a) to be a Young diagram, which is completely nonobvious from the definition of a poset’s shape.
References
Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.
Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.
Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.
Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.
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Gorodentsev, A.L. (2017). Calculus of Arrays, Tableaux, and Diagrams. In: Algebra II. Springer, Cham. https://doi.org/10.1007/978-3-319-50853-5_4
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