Standard Young Tableaux

Question

1. What are the contributions in this paper?

2. What is the quotient of a symmetric polynomial?

3. What is the common way to leave off explicit mention of variables?

4. What is the sign character of a given cycle?

Accepted Answer

Although the original motivation for the definition of these objects was the study of Macdonald polynomials and the representation theory of diagonal harmonics, in this account the authors focus only on the combinatorics associated to their description in terms of lattice paths.. Section 5 contains a brief account of the recent exciting extensions of these objects which have arisen in the study of string theory, knot invariants, and the Hilbert scheme from algebraic geometry.

Accepted Answer

The quotient ring Rn = C[x1, . . . , xn]/ < e1, e2, . . . , en >, or equivalently C[x1, . . . , xn]/ < p1, p2, . . . , pn >, obtained by forming the quotient by the ideal generated by all symmetric polynomials of positive degree, is known as the ring of coinvariants.

Accepted Answer

In a symmetric function it is typical to leave off explicit mention of the variables, with a set of variables being understood from context, so mλ = mλ(X).

Accepted Answer

χ1 n (β) = (−1)n−ℓ(β) for all β ⊢ n, so χ1 n is called the sign character, since (−1)n−ℓ(β) is the sign of any permutation of cycle type β.

Accepted Answer

(77)The definition of Cn(q, t) was motivated by ideas involving algebraic geometry, and Garsia and Haiman conjectured that Cn(q, t) ∈ N[q, t].

Accepted Answer

Thenen(XY ) = ∑λ∈Par(n)sλ(X)sλ′(Y ) (50)hn(XY ) = ∑λ∈Par(n)sλ(X)sλ(Y ). (51)There is a q-analogue of the Kostka numbers, denoted by Kλ,µ(q), which has many applications in representation theory and the combinatorics of tableaux.

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It follows that the pmaj parking order of CY C(P ) can be obtained by starting with the pmaj parking order β of P and performing the cyclic-shift operation on each element of β individually.

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Given a path π ∈ L − n,n, let P be the point with smallest x-coordinate among those lattice points (i, j) in π for which j − i is maximal, i.e. whose distance from the line y = x in a SE direction is maximal.

Accepted Answer

If this occurs at (k, k), then the area of the part of π from (0, 1) to (k − 1, k), when viewed as an element of L+k−1,k−1, is k − 1 less than the area of this portion of π when viewed as a path in L+n,n.

Accepted Answer

For 1 ≤ k ≤ n,zk = ∑n≥kq( n 2)−n+kFn,k(q −1, 1)zn(z)n. (132)Theorem 19 is a q-analogue of the general Lagrange inversion formula [AAR99, p.629]f(x) = f(0) +∞ ∑n=1xnn!φ(x)n[dn−1dxn−1 (f ′(x)φn(x))]x=0, (133)where φ and f are analytic in a neighborhood of 0, with φ(0) 6= 0. To see why, assume WLOG f(0) = 1, and set f(x) =∑∞ k=0 ckx k and φ = 1H(−x) = ∑∞ k=0 ekx k in (133) to get∞ ∑k=1ckx k =∞ ∑n=1xn n! H(−x)n dn−1 dxn−1(∞ ∑k=1kckx k−1(∞ ∑m=0emx m)n)|x=0 (134)= ∞ ∑n=1xn n! H(−x)nn ∑k=1kck(n − 1)!( ∞ ∑m=0emx m)n|xn−k (135)= ∞ ∑n=1xnH(−x)n n ∑k=1ck kn∑j1+j2+...+jn=n−k ji≥0ej1ej2 · · · ejn (136)= ∞ ∑n=1xnH(−x)n n ∑k=1ck kn∑α⊢n−keα(nk − ℓ(α), n1(α), n2(α), . . .).

Standard Young Tableaux

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Most frequently asked questions

1. What are the contributions in this paper?

2. What is the quotient of a symmetric polynomial?

3. What is the common way to leave off explicit mention of variables?

4. What is the sign character of a given cycle?

5. What was the motivation for the definition of Cn(q, t)?

6. What is the q-analogue of the Kostka numbers?

7. What is the pmaj parking order of CY C(P)?

8. What is the smallest x-coordinate in the path?

9. what is the area of the part of from (0, 1) to (k ?

10. what is the q-analogue of the general Lagrange inversion formula?

Figures

Citations

Hook formulas for skew shapes II. Combinatorial proofs and enumerative applications

Asymptotics of the number of standard Young tableaux of skew shape

Bounds on certain classes of Kronecker and q-binomial coefficients

Hook formulas for skew shapes III. Multivariate and product formulas

Hook formulas for skew shapes

References

The Symmetric Group

Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen.

A variational problem for random Young tableaux

Hook young diagrams with applications to combinatorics and to representations of Lie superalgebras

Asymptotics of Plancherel measures for symmetric groups

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