The Algebraic Combinatorics Seminar

Scheduled for Fridays at 3:30 PM, Room 210

In reverse-chronological order.

Date Speaker Title (click titles for abstract) 11 Nov. 2016 Mike Zabrocki Find the combinatorial rule for the branching rule between GL(n) and the symmetric group 4 Nov. 2016 Nantel Bergeron The theta map

(1) The start is the theta map for Sym (Symmetric functions) in Macdonald. Read Chap 3 Section 8 (its pretty much self contain if you read the first chapter). Look at Exercise 10, the map is there... but called $\varphi$.

(2) The map was extended by Stembridge to QSym. In his case he extended the theory of Stanley's P-partition to understand Q-Schur functions:

Enriched P-partitions. Trans. Amer. Math. Soc. 349 (1997), no. 2, 763-788.

See remark 3.2 in the above paper

(3) We [Steph-Stef-Frank-Nantel] show that this is a Hopf map and is related to Eulerian poset

Discrete Mathematics, Vol. 246, 1-3 (2002) 57-66.

JCT-A, Vol. 91, 1-2 (2000) 84-110.

(4) Now in this context, the peak algebra takes its full importance as it encode the generalized Dehn-Sommerville relations. The paper [Marcelo-Frank-Nantel] fully show the context and the importance of these algebra maps:

Compositio Mathematica, Vol. 142, 1 (2006) 1-30.

See example 4.9

(5) Now, to push this further, you may look at [Marcelo-Kathreen-Nantel]:

Trans. Amer. Math. Soc., Vol. 356, 7 (2004) 2781-2824.

Section 7.4 contains some better understanding of the tetha map and some extension to type B.

(6) These are the kind of map we failed to lift to the Malvenuto-Reutenauer (MR) algebra.

(a) The odd subalgebra of MR is NOT the homomorphic image of a morphism (as far as I know)

(b) As I remember (with Mike) we programed, "what are the Hopf morphism of MR that would commute with the other maps involves". If my memory serve me well the answer was "none".

So we now replace MR by V (Vargas. I am sure it is elsewhere too, but cannot find it):

http://www.dmtcs.org/pdfpapers/dmAT0172.pdf28 Nov. 2016 Nantel Bergeron computations in quantum Fomin-Kirrilov algebra. 21 Oct. 2016 Carolina Benedetti and Laura Colmenarejo Discussion on the quantum Bruhat network As a first step, we study the paper SCHUR TIMES SCHUBERT VIA THE FOMIN-KIRILLOV ALGEBRA in order to understand better the quatum pieri rule for Schubert calculus.14 Oct. 2016 Carolina Benedetti Discussion on the quantum Bruhat network Nantel plan to

- Justify the definition of power sum in quantum case

- pose the problem of understanding Quantum pier rule in term of increasing path in a quantum k-Bruhat orderâ

- sketch a second proof of Frank's result in term of paths only7 Oct. 2016 NO SEMINAR We go to Convexity workshop 30 Sep. 2016 Frank Sottile A Murnaghan-Nakayama rule for Schubert polynomials in quantum cohomology 23 Sep. 2016 Alex Woo List of problems related to Reduced Total order We have a nice list of problems related to the order R and the order weak(R)

-1- Find an injection from $Inv(w)$ to $Exc(w)\times Def(w)$

-2- Show that $L_R(w) = L_R( \gamma^{-1} w \gamma )$

-3- Show that $L_R(w) = \sum_{(a,b)\in Inv(w)} \chi_{w}(a,b)$ where $\chi_{w}(a,b)$ is zero when $a$ or $b$ are fixed point of $w$, is 1 if $a\lt w(a)$ and $b\gt w(b)$ and is -1 otherwise. The right hand side of this formula comes from the rank function in the [Bergeron-Sottile] (Grassmannian) order. see Section 3.2 of

Duke mathematical journal, Vol. 95-2 (1998) 373-423

-4- Show that the order of [Bergeron-Sottile] is the same as the weak(R) order.

This is shown by Bennett and Blok in Partial orders generalizing the weak order on Coxeter groups, JCT-A Vol. 102 (2003) 331-346

-5- Show that the maximal element of the order R are exactly the non-trivial power of the long cycle $(1, 2, ... , n)$.

-6- Can we say something about the interval $[Id,w]_R$ where $w$ is a maximal element of the R-order. (remark that for the same max element the interval $[I,w]_{weak(R)}$ are very nice, they should be the Young lattice in the rectagle defined by $w$.)

A simple SAGE program to compute the rank and Hilbert function of this order follow this link.

16 Sep. 2016 Alex Woo Understanding the "reduced absolute" order of the symmetric group The reference for reduced reflection length is p. 26 of Bagno-Biagioli-Novick-me:

http://arxiv.org/pdf/1507.01180.pdf

The only reasonable way I know to calculate it (for S_n) is to combine Proposition 6.6 (which is an equality for S_n) and Theorem 1.1 in Petersen-Tenner:

http://arxiv.org/pdf/1202.4765.pdf

As far as I know, the notion of "reduced absolute order" is not defined in print anywhere.

Some of the kinds of questions one might be interested in were answered for absolute order (the order based on reflection length) in

http://arxiv.org/pdf/0706.1405.pdf

and various papers it refers to. (For Bruhat order (the order based on length), these have been well known for a very long time.)

[Added by Nantel: It seems like it is very interesting to look at the "nil"-operators algebra of these poset as in Benedetti-Bergeron: http://arxiv.org/pdf/1307.2528.pdf

my initial computation reveals that it is very interesting in these cases]9 Sep. 2016 Nantel Bergeron Four Suggested problem for this year (Mike Zabrocki, Frank Sottile, Alex Woo and Nantel Bergeron). Mike: Find the combinatorial rule for the branching rule between GL(n) and the symmetric group click here to see moreTake an irreducible $Gl(n)$ representation and consider the subgroup of permutation matrices contained in the group of all invertible matrices. The goal of this problem will be to determine how this irreducible decomposes into permutation group irreducible representations. Representation theory gives one answer to this question, namely, find the Schur expansion of the Frobenius image of the character of the irreducible $Gl(n)$ representation. Since the irreducible $Gl(n)$ represenation has character equal to a Schur function $s_\lambda[X_n]$, we are asking for the Schur expansion of $$\sum_{\mu \vdash n} s_\lambda[\Xi_\mu] \frac{p_\mu}{z_\mu}$$ where $f[\Xi_\mu]$ means evaluate the symmetric function $f$ at the eigenvalues of the permutation matrix with cycle structure $\mu$.

A new approach to this problem is to use a tool of a basis of the symmetric functions which represents the irreducible characters of the symmetric group. The problem of finding the coefficient of $s_{(n-|\nu|,\nu)}$ in the equation above is equivalent (for $n>|\nu|+\nu_1$) to asking for the coefficient of ${\tilde s}_\nu$ in $s_\lambda$ where ${\tilde s}_\nu$ is the irreducible character basis of the symmetric group represented as permutation matrices (see reference [2]). We don't have a combinatorial interpretation for the transition between the Schur and the irreducible character basis, but we do between the homogeneous (complete) basis and the irreducible character basis and between the homogeneous and the Schur basis.

Essentially, we are trying to complete the following diagram.

The coefficient of $s_\lambda$ in $h_\mu$ is equal to the number of column strict tableaux of shape $\lambda$ and content $\mu$. The coefficient of ${\tilde s}_\lambda$ in $h_\mu$ is equal to the number of column strict multi-set valued tableaux of skew shape $(k,\lambda)/(\lambda_1)$ and content $\mu$. The column strictness of the multiset tableau is with respect to lexicographic order on the multi-sets...but technically you can choose any total order on multisets if it makes the problem easier to solve.

If you would like to compute data for this problem, some programs are already implemented in Sage.

sage: Sym = SymmetricFunctions(QQ)

sage: s = Sym.schur()

sage: st = Sym.irreducible_symmetric_group_character()

sage: st(s[2,1])

st[] + 3*st[1] + 2*st[1, 1] + 2*st[2] + st[2, 1]

sage: s[2,1].character_to_frobenius_image(5)

s[2, 2, 1] + 2*s[3, 1, 1] + 2*s[3, 2] + 3*s[4, 1] + s[5]

[1] K. Nishiyama, an unpublished paper about the restriction problem from $Gl(n)$ to $S_n$ and its connection to plethysm.

Restriction of the irreducible representations of $ GL_n $ to the symmetric group ${S}_n$

[2] R. Orellana and M. Zabrocki, Symmetric Group Characters as Symmetric Functions

[3] R.C. King, Branching rules for $GL(N) \supset \Sigma_m$ and the evaluation of inner plethysm

[4] This seminar from 2006-2007 and again in 2009-2010 and notes developed from that discussion.

Frank: Prove conjectured Murnaghan-Nakayama for quantum Schubert polynomials click here to see more[1] A. Morrison, A Murnaghan-Nakayama rule for Schubert polynomials, FPSAC 2014

[2] A. Morrison, F. Sottile, Proof for ordinary cohomology, Annals of Combinatorics

[3] K. Meszaros, G. Panova, A. Postnikov, Schur times Schubert via the Fomin-Kirillov algebra, Electronic J. Combinatorics, (2013).

[4] A. Postnikov, On a quantum version of Pieri's formula, Advances in geometry, Progr. Math., vol. 172, Birkhauser Boston, Boston, MA, 1999, pp. 371--383.

Alex: Understanding the "Reduce" order of the symmetric group

Nantel: Complete the square for the Theta map.

- Unless otherwise indicated, the seminar meets on Fridays at 3:30 pm in Room 210 of the Fields Institute.
- If you are interested in speaking at the seminar, contact Nantel Bergeron or Carolina Benedetti.
- You may also be interested in the Applied Algebra Seminar (Monday afternoons at York University).

** About the seminar. **
Every year we pick a new topics to explore.